My 2 cents.
You seem reasonable. The other parent? confrontational. So not legal advice. Just life experience.... My ex still pulls this BS. I'm not a solicitor and neither are you... BUT we've both been lurking on this site long enough to have learned a thing or two... Meanwhile my ex gets her advice from less reputable sources like mother's group, or her best friend's neighbours cousin who works at CSA....
When the ex tells you the'yre applying to court, or any other BS, you need to smile and nod... SMILE AND NOD....
Best therapy I have when the ex is causing my head to spin? I read my court orders. I read the bit that states "the children shall live with their father". I like reading that. It reminds me that an authority higher than my ex has determined that "the children shall live with their father"