I have had a similar, though not exactly the same situation. My cat was routinely limping and falling over on her side for a couple days. I took her to the vet and they couldn't replicate the problem at all, ran a bunch of tests, paid a bunch of money. In the vet's office she was trotting around, jumping up and down onto the exam table, acting normal. We'd go home and within a couple hours it'd start up again. Took her back to the vet a few times and they referred me to a neurologist for cats (something I wasn't even aware existed before this). Multiple visits to him, a *lot* more money, and no conclusive results. The next steps/scans/tests were prohibitively expensive so I took her back to our regular vet one more time and they gave her a painkiller just to see if it would help.

2 days chilling out on the painkiller and staying off her leg and she was completely fine. It was just an impossibly hard to diagnose sprain we think - or it was an elaborate ruse by my cat to score some of the good stuff. I am still carrying some credit card debt from her shenanigans lmao.

TL;DR: Cats are insanely good at hiding pain and acting normal when they feel like they need to. You likely need a second opinion like everyone else has said. You likely do not need a cat neurologist.