My poor love. That sounds awful. Absolutely horrendous. I wouldn't wish that level of sleep deprivation on anyone.
Don't think I've got any great advice, but hopefully it will bump the thread so others more knowledge come along.
Is there anything that makes him sleep during the day, pram, car seat? I used to walk miles during the day so he'd have a decent nap in the pram, that sometimes helped him sleep better at night.
Is he too hot or cold at night?
My son is about the same age and I've gone through long spells like that, it's brutal.
Things that have helped us (obviously all babies are different):
Massively overestimating him during the day, lots of new people, lots of exercise, lots of fresh air and daylight. We've discovered that a few days of intense activity (visiting family) has him crashing out with exhaustion.
Feeding him lots during the day
The odd dose of baby nurofen in case he's teething.
Occasional cosleeping when required.
Occasional dosing with gripe water before bed
White noise, sometimes all night
Blackout blinds
Olbas oil in the room in case he's stuffy
If I'm not feeding him to sleep, I end up cuddling him while rocking in a chair while he goes ballistic at me and he eventually goes to sleep, usually less than 15 minutes.
Do you have any family support? Can you get them to come round for a day or two, get them to take the kids out all day so you can just sleep. Might be enough to just take the edge off your exhaustion as you must be dead on your feet.
And look into getting a different gp, one that's a bit nicer.
I won't say "it'll pass" etc, cause when in the depths of sleep deprivation he'll I usually wanted to punch people who said that. I hope it starts to improve.