It becomes legally challenging with regards to ownership if I ever use work equipment for a personal project. If it later takes off they could very well try to claim ownership in its entirety simply because I ran a test once (yes, there's a while silicon valley season for it).

I don't know if they'd win, but Anthropic absolutely would be able to claim the creation of that code was done on their hardware. Obviously we aren't employees of theirs, though we are customers that very likely never read what we agreed to in a signup flow.