There is a great EU261 thread (sorry can’t link it as I’m on phone). Have a look in there and look to take it to CEDR. The process is detailed in the wiki of the thread. It sounds like you definitely have a case and they should pay out. But BA being BA are trying their luck and hoping this excuse will stop people from pursuing a claim.
I wish EU261 had a clause that found if the courier made false claims when denying EU261 then they’re fined heavily.
ps. It is for BA to prove it doesn’t fall within EU261. So have them send evidence.