So many of us have been in your shoes. So, you are absolutely right that your child can make it through the night in terms of hunger - our pedi told us that children make up during the day once they stop feeding at night, and that was when our son was 6 months old. You are not getting any sleep, but neither is your baby! He needs it. He needs to learn to self-soothe, starting now. Taking him into your bed leads to problems down the road, in my opinion - but in your case, it's not working anyway.
Our son did well with a white noise machine - we use one of those dust filters to get a double-use out of the purchase, but you can use anything. When he was little, he had a teddy bear that made a heartbeat sound - supposed to simulate the mother's heartbeat. Your son might be a little past that by now though. If he doesn't have a nightlight, you could use one, but I don't think that's what's going on.
We did not pick our son up (on our pedi's advice) nor did we feed him. We patted his back, said "Time for sleep" and just one or two other soothing phrases (you pick what works for you but just repeat those few things endlessly). We went in less and less frequently (10 minutes, then 15, then 25, etc.). Yes, it's a hassle, yes he cried, and no we didn't get much sleep for a few days. BUT no one was getting sleep up until that point anyway. We didn't feel it was cruel to have him cry it out because we weren't leaving him in there for hours with no attention - we just gave him the confidence to settle himself down, and he knew we loved him even if he wasn't getting what he wanted. (A good thing for him to internalize for his later years too!) Bottom line, he needs to learn to calm himself down and put himself to sleep - he can do this even though right now he doesn't seem to.
I would suggest that you choose a method you and your husband are comfortable with, and just DO it. Stay consistent no matter what. Maybe you could do this over the school vacation so your 5 year old isn't exhausted during school. Or have the 2 older kids stay with Grandma? Whatever works. But you cannot continue the way you are - it's not fair to the 10 month old even more than it's not fair to you and your husband. He can't grow, learn or have a decent relationship with his siblings if he is sleep-deprived.
Good luck - I know this is not easy but you will be glad you did it.
And by the way, I absolutely disagree with the suggestion to put baby powder in the diaper! Ask your pedi but ours told us it causes huge problems with yeast infections, and you do NOT need one more irritation in your child's life!