To answer the questions: I leave home 25 hours before the BA flight I need to check in for, so I can't print the boarding pass before I leave. I can use the app, but when I arrive at LHR I won't have mobile network data on my phone and I've had bad experience with trying to rely on wifi to let me access boarding passes on my phone.

Sounds like it's worth attempting at least -- I'm flying in business class so hopefully that will convince agents to be more cooperative. If that doesn't work I'll use the app and hope that I don't run into wifi problems.