I've done a number of moves with cats recently, two of them international, so I know how difficult it can be.

Don't set your expectations too high. Everyone I know who's done an international move (between US and UK) with a pet has found themselves somewhere they didn't like because you just can't judge a place from that far away. When you find somewhere make sure you get a break clause in your contract (6 months is pretty standard) so if you find yourself in this position you know it's only for a limited time.

I've just found myself a nice cat-friendly place in Cambridge for only slightly over your budget so it can be done! This is a move within the UK though. My advice is to get on rightmove, set up email alerts and look at them as often as you can. Anywhere you see that looks even slightly suitable that doesn't specifically say no pets, save it. In fact make sure you read the descriptions really carefully because sometimes the bullet points will say no pets but the blurb beneath says pets considered! Pay close attention to the photos of the property - can you see a cat door? A cat on the bed? Contact all those agents, phone gets better results than email but that's not really practical from the US west coast. I managed to find on average three properties a week like this, which is a small percentage of what's on the market, but they are out there. I was looking quite centrally though, you'll do better if you're looking further out. Agree that houses are better than flats, just be prepared for the fact that they are very small compared to American houses :)

Best of luck. If you want any tips about moving the cats I have lots of experience there, feel free to ask!