I have a lot of sympathy. I have a newborn too, and had four hours of non-stop cluster feeding the night before last. I feel lucky this morning because he only woke every hour instead of being constantly awake. I remember from last time being so tired that I hallucinated.
I agree heartily that the thing to look at is what else can you drop. Giving the baby over to your partner at some point so you can get some uninterrupted sleep is so important. It's quite right that a few odd hours without milk won't be terrible for the baby, and you might also leave a few mls in syringes or a small bottle.
Do as little else as possible right now. Go to bed as early as you can, get up as late as you can. This is a relatively short, intense, phase - and you should focus on putting yourself first; eat well (or at least, eat frequently, including overnight snacks) and drink lots.
If you do want to use formula occasionally, it's not the end of days. I started out mix-feeding my eldest because he'd been formula fed in NICU for five days, and my supply took a few days to catch up. I fed and fed, but during second week growth spurt, I was wiped and desperate, so we gave him some ready-made formula, and nothing terrible happened, and he (briefly) stopped yelling. I also felt slightly calmer that we'd managed to fill him up. We did this only a few times, and it was fine - no effect on supply, no confusion over teats (although bottle feeding wasn't something he liked, I think I'd syringe feed if I was in that position again).
Also, breastfeeding support groups are wonderful. Everyone is in the same boat, or has a clear memory of what it's like, and there'll be tea and biscuits. I trained as a peer supporter after attending them with my eldest child, because they were just so valuable as a supportive space.