I know you're not going to like this answer, but there is absolutely NOTHING you should do to get any governmental employee off your back when a false allegation has been made against you.
The reasons any governmental employee looks into allegations of any kind (false, true, inconclusive, etc.) are simply because the governmental employee does not wish to lose his/her job. If they actually cared one whit about the facts, they would actually investigate the tipsters for ulterior motives prior to simply taking action based on the belief of the tip(s).
The fact of the matter is this: they aren't paid to give a hoot about anybody, but they ARE paid to APPEAR to give such a hoot, and the EASIEST way for them to appear to give as much of a hoot about as many of the citizens under their supposed jurisdiction is for them to swoop in and make the most drastic action possible at the very outset (make an arrest, take you into custody, take your children away, write a protective order, etc.).. and then... based upon their availability (in other words, based upon the amount of on-the-clock time that they are NOT involved in more-important aspects of other cases) they will FOLLOW UP, just as long as they have to, to make it APPEAR as if they still give a hoot about what is going on, lest their higher-ups decide to cut them from the team for not appearing to care about things that they should appear to care about.
So, in order to keep on APPEARING to give a hoot, they will do whatever they can to drag your case out as long as they can get away with it... because if they stop appearing to give a hoot, they will have to drop the case, which means they will have a decreased caseload, which might also mean less justification for their continued employment.
So, the only TRUE answer to your question, would be to find a way to cause each governmental employee involved to understand that it would be much more lucrative for them in the long run to stop giving a hoot about your case... HOWEVER, would be considered a BRIBE.. which is always illegal, and is almost always considered a crime... which is only ONE of the reasons why I don't even BEGIN to recommend that method, another being the appearance of guilt.
If you were to, in any way, shape, or form, modify your behavior during the so-called 'investigation' (see above), that would give them just that much more reason to EXTEND their period of appearing to care... under the justification that if you are behaving differently now that you're being investigated, then you really MIGHT have been doing the very things that had been alleged... and they would use that as an excuse to keep propping up their caseload with your still-open case.
If you were to, as you say, "prove it's false", that would be modifying your behavior, so, even if you WERE to prove it's false, they would STILL keep your case open, under the exact same justification... that maybe you WERE doing it... or something extremely like it.. so even if THESE allegations were proven false, it doesn't mean that they couldn't find something SIMILAR in your behavior, once your behavior returns to what your neighbors tell them is 'normal', which, of course, could wind up being a VERY VERY VERY long time, during which time, of course, they would continue to be 'on your back' about it.
Now, unfortunately, when I say there is 'nothing you can do', I do NOT mean that you should do nothing when you could do something... what I mean by that, is that even though there is absolutely zero legal or lawful obligation for you to answer any of their questions.. REFUSING to answer any question at all, will give them as much room as they so desire to pick and choos as many of your neighbors' comments to help justify the perpetuation of their case... however... ANSWERING questions gives them the opportunity to do whatever they so desire with your questions.. and for a LOT of governmental employees.. they will do a LOT of paraphrasing in their reports... what I mean by that is that if they were to ask you the question "So, why do you think I'm here?" and you were to answer "Somebody lied to you about my misconduct towards my children" they wouldn't HAVE to put anything like "He thinks somebody lied about misconduct", but they COULD (legally, lawfully, ethically, etc.) put "Subject confessed misconducted" simply because of your use of the phrase "my misconduct".
Furhter, even if you were to have separated those two words like thus: "Somebody lied to you about my actions [blah blah blah blah blah blah blah blah] and I don't even begin to understand how this could ammount to misconduct", the rules of "ethics" allow them to paraphrase your comment thus "Somebody lied to you about my [...] misconduct" in their minds and then put "Subject confessed to misconduct" in their report."
So, they've got you between a rock and a hard place, whatever you do.
And, again, there is absolutely NOTHING (short of successfully bribing all those involved, which I do NOT recommend), which will get them off your backs.
It is only until THEY decide to drop the case due to unsubstantiation of allegations that THEY will get off your back.. and not a moment before then.
That being said... I DO have a very powerful tactic to use to get them to hurry their usually lazy behinds up about getting the allegations substantiated or not. I've used this tactic dozens of times with the exact same results in all but three instances.
When they get around to the point in their "investigation" where they begin to ask you questions... simply respond to EACH AND EVERY question with the sentence
"I choose to defer providing an answer to that statement until the trial, thank you" and say absolutely NOTHING more about it.
Every single time this has occurred in my life, the interrogater has ALWAYS started off pretending to be kind and polite and caring... every time I have answered quesitons, they would continue to pretend to be kind and polite and caring, and then they would put in their report that I had confessed to things that were untrue.
Every single time that I've responded with nothing more than what I recommended above, ALL of them pretended to be polite, courteous, and/or caring for several more questions... but, EACH of them.. at some point before the tenth or twelfth question began to let their true colors show, and start playing mean, by threatenting me with all sorts of things, like "If you don't answer, I'm going to take your kids away from you forever" or "..., you're going to be locked up for a very long time" or "..., the fines you're going to have to pay are going to force you into bankruptcy" and a whole lot more threats.
At that point, I simply say "Reporting ANYTHING that I did NOT put into my answer will result in me placing you under citizen's arrest for the falsification of government documentation, which, even if you DON'T wind up being convicted of it in a court of law, WILL result in getting this case given to someone else in your office, and when there are no more of your kind to investigate me, you will HAVE to drop the case, which I happen to know will be HEAVILY frowned upon by your supervisors, and might actually add you to the rolls of the unemployed, so, I wouldn't even BEGIN to TRY to threaten me with things that could put your career and your freedom in jeoboardy, if I were you, and all you have to do to find out how serious I am about it is to try it, so, I am going to declare right now that this interrogation is at an end, unless you have some other relevant questions to ask of me."
In none of my cases was that the end of the interrogation, but in most points, there were more questions which I deferred until the trial.
In some cases, the interrogator would TRY to threaten me more, at which point, I would stand up and step towards the door, before getting asked another unanswered question.
In some cases, the interrogator would say "There isn't going to BE a trial" or "This case isn't going to court" or something similar, to which I would respoind "You have no idea how glad I am to hear that when I file my lawsuit, you're going to testify that you didn't have enough evidence in the first place to merit taking me to court over it, and it's quite clear that you don't have enough so-called evidence without somehow revising my answers to be used against me, which I flat out refuse to let you do, or you would be going ahead with your case instead of still investigationg it. I already know where you work, but, I don't necessarily want to serve you your subpoena in front of your boss or your co-workers, because they might not like to hear that you're going to be providing testimony against your whole department and therefore against them. I'm willing to keep it hush hush, if you're willing to give me your home address or your attorney's address."
In some cases, the interrogator would start asking me whiny questions about "How am I supposed to do my job if you won't answer my questions?" or something like that, which has given me a LOT of fun. Once my response was "The word 'supervisor' comes to mind.. but then, so do the words 'negative performance review', 'termination', and 'unimployment', so I can easily understand why it is that you are reluctant to confide your incompetency in your supervisor." Another time, my response was to get up, reach for the phone, ask me for her supervisor's business card, and begin to dial that number.
My record for getting them off my back speedily (and the most recent of such incidences) occurred when my response was "Well, since you're obviously too concerned with your supervisor learning just how grossly incompetent you are in your chosen path of employment to get him or her to actually TRAIN you in how to do your job competently, you might try this: instead of seeing if you could trick your subject into giving you an answer that you could twist around to suit your own career goals, you could always try gleaning as much useful evidence and/or testimony from the liars who got this big huge snowball rolling, and see if you can't actually learn that there really IS no evidence supporting their false claims."
And when she asked me ANYTHING ELSE at all, I simply said "I've given you ALL you need to know to begin doing your job.. so do it., and leave me alone" like a broken record.
That particular case was closed after one single phone call, about fifteen minutes after the twit decided to stop trying to interrogate me, because even at her diminished capacity to understand things, she, too, was able to learn that the reports were all lies.
The only time this tactic hasn't worked was in a town in Utah where the judge was the only competent prosecutor, so he would wind up prosecuting each and every case while the imbecilic prosecuting attorneys did very little more than keep their mouths shut for the whole process.... which of course defeats the whole purpose of having a court of law in the first place... that case is still unresolved, since the biassed judge knows that the moment he states there to be any kind of penalty at all, whatsoever, I'm going to appeal, and thereby expose his injustices as far and wide as I possibly can, which, of course, he does not want to have happen... so, he simply refuses to issue a "sentence" so, even my tactic doesn't always work out.