N.B.
You could try swaddling him and putting an infant seat or his carseat in the crib to sleep in so he's propped up. This worked wonders for my niece and she slept in it(still swaddled) until 5-6 months. Also, infant simethicone drops for gas and reflux work really well if you haven't tried them yet. Good luck, it won't last forever. As for you, nap when he naps, day or night. The older ones should be in school or still napping. Use gates or locks so they can't get into any trouble while you're resting. Everything else can wait, or be done by someone else! Don't be afraid to ask for help from a friend or family member. Dad's getting plenty of sleep, so he needs to step up and fix meals, bathe the older ones,and amuse them so you can get some rest before he goes to bed. He's a partner, not just a breadwinner.