I did this a few weeks ago. Arrived at T8 around midnight, made my way back in with my 6 am (for LAX) boarding pass. Went to the Greenwich lounge until it closed at 1.15 am. Found a seat at some gate, got comfortable with Netflix/Spotify/study until 4.15 am when Soho opened. Was the first guest in the Soho lounge, took a shower and breakfast. To the gate at 5 am.

All very civilized and comfortable. Some other pax and workers around. There is constant music in the terminal (regular pop, not too loud) plus announcements that you are not supposed to take firearms through security.

My plan B was a daypass for the gym (25 USD) at the TWA to work out a bit and have a coffee/snack there. Plan C was getting into Terminal 4.

I did manage to catch some Z's on both LHR-JFK and JFK-LAX flights. Arrived in LAX refreshed and rested.