Anecdotally most of those I've known recently who have traveled to the UK or transited on passports that require an ETA have not been asked for proof of it by the airlines, so it's highly likely that the airlines might have access to ETA and BRP info. I know it's a lot easier to prove having an ETA via an email though I don't know if there's anything similar like that for a BRP. They haven't posted explicit guidance about this though if I were in your situation I would contact UK Visas and Immigration about this as soon as possible to know what their guidance is for airlines and know what to submit to them as proof of BRP if requested. If you do contact them and get any info please let us know so everyone else here may benefit, too.