I'm really sorry that I've only just seen this thread, you sounded on the verge of pnd - but clearly you weren't there yet because you could see the need to ask for support. This is positive.
I wanted to post because it sounds a little like the problem I had with my daughter. She would eat, pull off and fall asleep and 10 mins later she would eat again. I couldn't do anything and was just a feeding machine. I hated it.
Don't beat yourself up about giving a bottle. I got through without giving a bottle and am quite militant about breast feeding but getting beyond that line and falling into depression would do more harm to DC than the odd formula feed.
Before you do the bottle thing though you might want to try this - it's what worked for me. When DC is feeding keep a close eye and as they look tired pull their ear gently to wake them up (or any other way you can keep them with you). Or if not falling asleep then after they've pulled off, take their head firmly and put it back. This is harder because it feels like you're being cruel - you're not. They're tired but they need to eat so that they can sleep better. Obviously you need to use sensitivity and care, I'm not suggesting hurting the child - before the mumsnet police arrive.
The point is to get DC to take more and better quality milk so that it stays in the tummy longer and then they can sleep. (It doesn't sound like cluster feeding to me but snacking which means they take the watery thirst quenching bit and not the more nutritious stuff).
You may find that there is the difficulty of not producing enough though because that's the pattern your breasts have got in. So you might find it worth trying a bottle in the day to build up the supply for the next feed and then trying what I suggest. Hopefully that will lead to a bigger feed and a longer gap - meaning a bigger feed next time. The gap should get steadily bigger and the vicious circle will reverse into a positive one.
Oh and if you do the mixed feeding, don't forget you could start expressing in the day and then use that as some of the bottle feed.
What ever you did or try good luck! Both mine were/are counter-intuitive as babies - pulling away as a sign of hunger being just one example!