Here's my approach, and it almost doesn't matter that it's LGW to LHR. When I'm planning anything that mixes a forced late bedtime with a deadline the following day, I start by counting backwards.

In this case, a flight to the US means you need to be at Heathrow no later than 10 a.m. Transfer between the two airports might be an hour at midnight, but it will be longer at that time Monday morning, meaning 8:30 and maybe 8 as a potential departure time from LGW if you stay there overnight.

The remaining factors may differ between you and me. Are you a morning or late-night person? How long does it take you to get out of bed and out the door in the morning? How much risk are you willing to assume? And on this specific flight, how much could the BA lounges help you, especially with a shower and breakfast?

For me, the answers to these questions mean I stay as close to my arrival gate at LGW. Maybe for you, it means you should stay as close to your departure gate at LHR.