OP...how long is the flight? Guessing 6 hours plus if you're looking into British Airways. We almost always fly United as we fly out of Newark, but my strategy is to always set google alerts on the seats I'm willing to accept, and the seats I want. And, I've had great luck doing something most of us never do any longer....by picking up the phone and speaking to an agent.
We're flying to Ireland this summer on a red eye. It's only a 6.5 hour flight, but we'd like to at least be able to take a nap on the way over, and that's not possible for either of us in economy. We had generous flight credits ($2,500) from a cancelled flight from South Africa in Polaris, so we used those credits to get us seats in economy (not basic economy...as those seats can not be changed). I set google alerts for Premium Plus (which is the same as a United Business seat domestic), and Polaris. PP was showing RT at $6,000 which felt too high for this flight, but we were willing to pay $4,000. I felt confident we'd at least get PP for the price we wanted.
I got a Google alert that the price had dropped down to $4,200 RT for both of us....Polaris was showing $10,000 RT for the two of us (too high). I'd had luck calling before to see if an agent could do better....she was showing that price I was seeing for Premium Plus, but then mentioned for an additional $100 per person, each way (so $400 RT)....we could get Polaris, and so I pounced. $4,600 RT for Polaris these days is a good deal on any red eye to Europe...and we'll enjoy the Polaris lounge on our way out.
At the same time I was on the line with her, I was still seeing over $10,000 RT for the two of us in Polaris. People on the United sub reddit always say that we are not able to see all of the seats/prices available, but agents can. So...I've had luck doing that. Hope this helps.