I don't think it's going to be a problem flying with your ticketed names the wrong way round, just so long the name match your government issued ID such as passport or driving licence.

I've flown numerous times with my names messed up on international trips and you're only flying UK domestic anyway, you'll be fine.

Edit: If you're worried about lounge access, take your BAEC card with you. It is unlikely the TPs/Avios will post properly, if they don't, just follow up with a webform to BAEC after the trip and explain why.