Iām currently doing the same install in a 2 door.
It looks like JL makes the original carpeted stealthbox with 5 way binding posts and an uncarpeted version that is plug-n-play.
SB-F-BRONCG6/10TW1
SBX-F-BRONCG6/10TW1
This thread has a link to the female connector on eBay.
It looks like...
I had the same thing happen yesterday (lux trim). Screen worked when I started the vehicle with key fob and then start button. Drove 2-3 minutes and screen shut off. All button controls still worked. Screen came back on after ~15 minutes.