YWNBTA.. No way are you in the wrong here.
1) Your cat with paperwork to prove it, 2) super shady disappearance, 3) the poor kitty ran in fear as soon as he saw them.
I get a bit of a pit in my stomach sometimes because I've seen various autistic ESA's horribly mistreated and expected to just take the abuse, moreso than with neurotypical abusers who are more likely to be called out on it. Yanking at them, hitting them, squishing them, dragging them... Surely many are great caregivers but knee-jerk judgement on O'Malley's reaction? Unlikely in this case. Her condition is not a reason to back down or an excuse for anything here.

Take your dad's support and fight for your family member. (I can't believe your mom is so heartless about this honestly.)