mummyrunner congratulations on your baby. Is he your first?
The best advice I was given about the whole baby thing was my doula saying to me 'Give it 6-8 weeks'. And even then, it actually took 12 weeks before I felt like things were vaguely OK, and manageable, and I hadn't just made the worst mistake of my life. You are 5 weeks in to the hardest, hardest part; you're right in the thick of things right now. So I am so sorry to wind you up by repeating it but it will get better and you are nearly at the point when that will happen.
The other best advice was - the first 3 months are for free. Do whatever you need to to survive.
I echo everyone who says try a bottle now, whether expressed or formula. You need to do whatever you need to to get through this bit, and it sounds like you do know what you need. This is great timing for trying a bottle, isn't it? They say wait 5/6 weeks so there is no nipple confusion, and if you start now then you won't battle later.
The other thing is - can you bring yourself to try just giving in? My worry is that even with the bottle-feeding giving you a much-needed break, your baby is probably still going to cry a lot, and need you pinned to him a lot, and there's not much anyone can do about that, except put him in a sling and go out, or lie on a sofa watching every single episode of Frasier, according to your personality type. Sorry, that's not a helpful thought, but it may be the reality for the next couple of weeks.
Hang in there. It's not for long. Good luck OP.