Sleep when the baby sleeps doesn't work for a lot of babies. Didn't for my first. You have my sympathy OP but you know YABU.
DS slept for 45 min blocks overnight for nearly 6 months. Took about 45 mins to resettle after an hour long breast feed and then the cycle repeated. He napped for a maximum of 20 minutes until he was about 18 months old, when he finally learnt to do a glorious 1.5 hour stretch.
Trust me, I know how tired you are.
It was an absolute revelation when DD came along and slept for 3 hour stretches from birth. Thought there was something wrong with her!
My advice fwiw is get out every day, the more natural daylight the better. Cluster feed during the evening, if he's falling asleep during a feed, wake him. Don't feed at night more than every 3 hours and extend that to 4 after a few nights. Keep lights low and don't interact. Do whatever it takes to get him to sleep - white noise, cuddles, shh pat, but dont leave him to scream.
Find someone, anyone you trust, to give you a break. Even if it's not family. Do you truly have no one within 50 miles who would give you a break? Or speak to your HV as some areas have special setups of volunteers to give mums a break and help a bit.
Also, controversially, I would consider early weaning. With hindsight DS was underfed, I know that now having had a well fed one! He was born skinny, jumped up to 50 centile, but after weaning shot up to 91-97th. He's very tall and skinny still. I also think, he was refluxy and weaning s settle too.
If it helps, DS started sleeping through from about 8 months and is generally a fab sleeper still. DD who started so well continued to wake every 3 hours for 3+ years and still wakes most nights at least once at nearly 4. So your DS may surprise you yet!
Don't shut the door on him. You will regret that more than you can know.