Answering the first question first ... ground transfer from LGW to LHR can take an hour [or three]. The road system is vulnerable to traffic chaos. We have to do that all the time, connecting to the TATL to IAD. Your 4.5 hours "should" be OK, but we prefer to avoid the stress, and over-night at LHR for peace of mind!

As to the other options, if you can fly direct into LHR it will significantly enhance proceedings. Regardless of which Terminal you arrive at, it has to be better than the "Lottery" of an LGW-LHR ground transfer via the M-25.

If you can book straight through on BA from BCN, you do get the TATL baggage allowance on your connecting flight. [Does that apply to partner carriers?]

Why Ezy when you can do it easy?

Enjoy flying BA, whichever way you end up doing it.