I think it's time to get tough mummy out - they're not tiny babies that need round the clock feeding. As long as he's getting enough solids in the daytime he should be able to sleep through the night (or at least 6/7 hours) without bf - does he have anything else as a comfort for if/when he wakes? C has the music and a fleecy bunny blanket.
Is he in a separate room? I found C suddenly slept through the very night we moved him.
Do you get up at the earliest peep, or wait till full on wails? What do you try to get him back off before you offer bf - it should be last resort. What did you do with DS1 and 2? If music doesn't work with C - like last week he was teething and woke up through the night without going back to sleep with music, I'd go in and gently shush him, stroking his face, singing the lullaby, then I'd leave the room and leave him 15 minutes.
He barely fed because it's not nutrition or milk he wants - just boob 24/7, it's the same as babies who need to have their dummy in all the time. Bad habit!
Lots of sympathy for you though, you're just trying to be the best mummy you can be, and that's admirable!