Given her age, it does sound like the 4 month sleep regression. How long has it being going on? When DD1 went through it the every 45-minute wakings only lasted around 10 days, the rest of the time was every 2 hours. It did pass but I sympathise with you as at the time it felt like it never would!
I think you really are going to have to sleep in shifts to keep your sanity. If, as you say, she's not hungry then just give her to your DH for a bit so you can get a couple of hours in (that's assuming you have a downstairs or somewhere enough away from you that you can sleep through it).
What about using a dummy? On DD1's worst nights she would only stay asleep if my nipple was in her mouth. I'd always been anti-dummy with my own baby, but if I could have got her to accept my emergency one back then, I'd have had no problem with it! A few nights is not going to create a forever habit.
How is the sleep during the day? The worse the day naps, the worse the sleep regression gets as they are then overtired to boot. I used to walk DD1 for her first nap (9am) as I found it would make her sleep longer, then the following naps would go easier too (and so too the nighttime).
Remember that cosleeping doesn't work for all babies (certainly not mine or many friends' babies I know of) so don't feel bad if it's not going well. You're doing great!