If the flight was under codeshare, so AA1234 sold as BA9123 or similar, then EC261 would apply. If it was in AA prime then it probably still applies. Normally the operating carrier, AA, has that responsibility but BA will become, under CJEU jurisprudence, the fallback airline if AA refuses to pay. One drastic approach would be to MCOL both BA and AA as joint defendants, which will then be paid very quickly, but I'm sure you are trying to avoid that. My advice would be to submit the claim to BA, noting that AA had refused to pay, and claiming under "CS and others versus České aerolinie a.s. (case C-502-18)". After 42 days if a codeshare go to CEDR, if not a codeshare go to MCOL.

See the main EC261 thread for guidance on CEDR and MCOL.