I think you will struggle to upgrade the transcon to F. There seems to be some confusion here, but you can only use avios to upgrade AA flights if you have: a.) AA prime flight number on a BA 125 ticket and b.) J, C or D fare class.

It will also be hard to upgrade even with cash. You definitely can't do that through BA. I tried to do what you are attemting at JFK a while back and got told by 1.) the lounge, 2.) customer service desk and c.) the gate that you can't upgrade a BA 125 ticket on AA with cash (I ended up getting opup'd, would you beleive it!) I did hear one guy on another thread say he was able to upgrade an AA LHR-PHL on a BA ticket with cash at LHR; not sure if that was with an AA prime or BA codeshare and also not sure if that makes any difference. Maybe it's just that agents in LHR know how to do this; maybe just got lucky...

You'll be fine in business, and the Chelsea lounge isn't worth blowing $300-400 on anyway.

If you can somehow upgrade your LHR-JFK or JFK-LHR to F, then you would get access to Chelsea through that.