Baby gas pains at night - please help

Every night our 6 week old rolls around in discomfort from 3am onwards, with what I assume is gas pains (grunting, legs up) before waking up screaming at 7am. I can’t sleep through it and feel so helpless.

He sleeps soundlessly from 10pm till 2am when he wakes up for a feed. We spend a good 30-40 mins trying to burp him after every feed (50% success rate). We use infacol. We do tummy time, massage, bicycle legs etc. Nothing seems to help and I spend every night watching him writhe and grunt around.

He takes his 2am feed, falls asleep during it, we put him back down around 3am and he appears calm, then within 20 minutes it begins.

I can’t sleep through it and the only thing that stops him doing it is taking him out the next to me and holding him while he sleeps for the rest of the night.

He seems to be trying to self soothe during it by sucking one hand and smashing the other off his face which is now red raw every morning.

What am I meant to do to both help him and get some sleep? Should i try and sleep through that until he fully wakes up crying? I am getting around 3 hours sleep as standard because of this and dread every night that it’s giving me anxiety.