Historical question as BEAST is mitigated in TLS 1.1 and earlier TLS is deprecated.
BEAST is a chosen plaintext attack, possible in web browsers because cross-origin requests have cookies automatically attached. This puts both attacker-controlled data and sensitive data in the same encryption context, making chosen plaintext attacks possible.
If an application uses the authorisation header to transmit the session token, instead of cookies, then tokens are not automatically attached and such an attack would apparently (to my eyes) not be possible.
I understand we'd still want to migrate to newer versions of TLS, but this would have been a reasonable "sticking plaster" mitigation. Is there any reason we did not see this at the time?