I re-read your question and I assumed the lounge was land-side. I can't answer about being able to exit terminal 3 for a second time. However, isn't there a land-side shuttle connecting T3 and T5?

You can also check if the carriers have interlining agreements. That way, you won't have to claim your luggage at all in LHR. I suggest you keep things simple and just buy a pass to the BA lounges, if they permit.