Since you have to "enter" the U.K. in order to transfer to LHR, you will have to retrieve your bags at LGW and take them through customs. I personally would not trust an airport to airport transfer of luggage - too many things could go wrong.

I would take the express bus to LHR, with your luggage, and check your luggage in at the BA counter, take the tube into London and return to LHR at your leisure.