Sympathies here, but such doctor's note are so easy to get (paying a private GP will still be cheaper than paying for an upgrade) so there is no chance BA will upgrade you, and I unfortunately need to say rightly so.

Split up the trip and/ or ask your employer to fly in Business. You are definitely important enough for them to send you far away, so if they don't make suitable arrangements for you, you should IMHO consider leaving your role.