OP, it sounds like you are in desperate need of help. I went through a full 7 nights of getting no sleep due to my DH's snoring, and found myself hovering over him on the 7th night, with a pillow held in both hands, about to suffocate him! I think I shouted, something like 'I can't stand this fucking noise for another moment, I've had enough', which thankfully woke him up, otherwise I might still have been serving a prison sentence. People laugh when I tell them about it now, but I was absolutely serious, and think that unless you've been through it yourself, you can't fully understand how awful this can be. So I have a few suggestions, that hopefully might help or give you some other ideas of how to cope.
First of all, do you have any friends or relatives nearby where he could sleep for a few days, so that you can get some proper sleep, even if you do need to be 'on call' for your child?
Secondly, because you can't wear ear plugs, I would suggest going to your local pharmacy and seeing what they can suggest. There are various appliances that can be bought over the counter, which may or may not help. Obviously I don't know what you may have tried already.
Does he snore in a particular position? If so, and he wears PJ's then try the trick of a tennis ball attached to the PJ jacket, if he doesn't wear them, make him wear a T shirt, with the tennis ball instead.
Your DH clearly needs to lose weight, and if he's the one snoring he needs to be the one sleeping on the couch or whatever, as what sleep you DO get, needs to be good quality.
Other than that, phone your GP, and tell the receptionist that you need an EMERGENCY appointment, and when they ask why, tell them that it's private and you'd prefer to talk to the doctor as it's embarrassing, that should shut them up. I think you should both attend, and explain to your GP that you're at your absolute wits end, and fear that you might actually kill him, if you don't get some sleep, as failure to get good sleep for a continuous period, is actually a form of torture.
I DO hope you can get some good sleep soon, but if you can't sleep, then make sure you frequently wake him up, so that he doesn't either, at least that way, he'll have some understanding of what you're going through!