The irony is, I was in that position when RDR came out back in 2010; my son was about a month old when it came out, and he wouldn't sleep unless someone was holding him.
So I did what any self-respecting gamer would do: I grabbed the boppy and propped him up on my lap at an angle, dropped the volume to the lowest setting where I could still hear it, and would play for 4-6 hours at night while my wife slept. It was a great system--he slept like a champ, and on nights when I didn't game, I'd binge watch The Office, How I Met Your Mother, or one of any number of shows.
Good luck to you, you'll need it. Don't be afraid to ask for help, especially during those first six months.