Is there somewhere I can transfer the domain where they won't care? And should I host it somewhere offshore?
Transfer to a Chinese host and registrar. They generally don't give a fuck about US law.
I am transferring my Domain to NameCheap anyway, and I have offshore hosting that ignores DMCA. Not sure if there's much else I can do, but it doesn't seem like Apple really pursues it since sites like AppleInsider have been up for so long.
If it's a .com, .net, .org or any other TLD controlled by ICANN it doesn't matter where you transfer it to, any registrar that answers to ICANN will hand over your real info if served and the domain in a winning UDRP or WIPO dispute in a heartbeat.