Seeking custody is a major change to the custody and parenting, many states have a process
Change of Circumstances which can take a year or more. Did you ever have a Custody Evaluation? If not, then one may be necessary. You can cast his countermove as "sour grapes" trying to not limit contact with you since he didn't make the allegation until after your motion. I don't know how well that would work.
How old is your daughter? Generally courts don't like parents to film children like that. It far too easy for the parent to influence the child. Did he inform the local children's protection agency so they could file a report or investigate? Or did he merely file with the court. Your lawyer could state that if he was concerned for the child's welfare then he would have done something like that. Likely the court will not trust such a recording. Since this may get some attention in court, he may feel emboldened to make more such claims in the future.
My story... .I recall a series of events just before our scheduled divorce trial. School called and said they couldn't reach mother and so wanted me to schedule a visit with the pediatrician's office regarding vaccinations. When I later called and told my then-stbEx she raged at me, then unknown to me she raged at the pediatrician's staff. According to their medical notes he promptly "fired her" but I didn't immediately receive a copy of their "Withdrawal of Services" since she had temp custody during the divorce. Well, she knew she would look bad so she tried to make me look worse than her. I suspect that's what your ex is doing, trying to make you look worse than him plus trying to get one up on you. Usually courts are very reluctant to grant one parent sole custody, it took me 2 years in divorce and 3 years post-divorce to get full custody, for solid reasons.
My son was no doubt pressured by his mother to tell ER staff that I had beat him (on his shins of all places). This was reported a week after she raged at the pediatrician's staff and got herself "fired" (Withdrawal of Services), so my conclusion is that she had to make me look worse than her. Even the documentation stated that his bruises were typical for an active boy his age and in typical locations (his shins). She had been bringing him in for her allegations for over a year and a half and apparently he finally echoed her. But when the CPS investigators spoke with him later at a presumed "neutral" location, his school, he didn't support what was alleged during her ER visit. Since he was only 5 then, I searched for and obtained an out-of-print Clifford the Big Red Dog book, "T-Bone Tells the Truth". T-Bone felt much better after he told who really tumbled Cleo's bird bath.
The moral: Even if it hurts or is hard to speak up, it is better to tell the truth rather than lie or let others believe a lie. And the person will feel relieved as well.
