Hi OP. We have had an almost identical situation, DD is 5 months now and we gave in and got a sleep consultant two weeks ago.
Her first step was to ask me to keep a diary of everything DD was doing, so times in the day she played, nappies, naps, how she fed, etc, and to keep a food diary for me. After a week she came back and said she thought there had to be a strong medical component based on how many times she vomited, and that my diet was pretty awful (as you'll know, so little sleep for months is hellish and we were relying on oven pizzas etc)
I cut out dairy and eggs completely and ate v traditional bland food- meat, potato, vegetables, soup, brown bread. After a week of that the vomiting was hugely reduced but not gone and the sleep was no better.
She came to see us to go through a proposed timetable, suggested timings for feeds and activities and naps and a plan for gradual withdrawal but no leaving her to cry. But after watching her she said that she had terrible silent reflux and nothing behavioural would work until we sorted it.
She doesn't scream and cry, she just doesn't really sleep for more than 40 min, an hour tops. I didn't notice back arching but the sleep consultant who has worked with children for thirty years did. She also said she was making little faces because of the acid taste. I increasingly don't think it's possible for children to have that little sleep so young without a medical issue.
For us, I think it's a combination of a food sensitivity and reflux. We're back to the GP this week. I have been recommended baby probiotics and some people I know have used osteopaths, not tried yet. I've cut out dairy soy and eggs from my diet. If all else fails I've been told when we wean her it will likely get better.
One thing I will say is that her daytime sleep is no better really- she'll only sleep in motion in the pram, being held in the sling by her daddy or sometimes in the car. I think the fact you're getting daytime sleep is a bit unusual. There might be something in that, but you're too tired to be expected to figure that out solo.
Also, as she got older it has got marginally better in the sense of getting the odd two hour sleep and her 24 sleep totals (I use an app to track all sleep) getting higher.
Don't sleep train, it won't work and will be distressing for you all. Everyone says it doesn't work till 6 months and to be honest, it just won't be behavioural/habit at that age.
I also have an oversupply but DD seems better at controlling it, that might be something to see a lactation consultant about it you haven't- I did block feeding and partially reclined feeding at it's worse and that helped.
I understand what it's like to have tried literally everything, I sat up at night downloading sleep books and nobody else's baby seemed to be as 'bad'. All I'll say is, it can be really hard to fix reflux, but what you're describing isn't normal and almost certainly is a baby in some form of discomfort and it's about trying to fix that, and in the meantime survive. Do you have a DP who can help? Is there anything that works at night?
We found one of us holding her upright or me feeding got us some nighttime sleep/quiet so we slept in turns. If you're just getting screaming that's so so hard. All I can say is do what you can to survive, and that if you can find someone v good with children and recommended like a sleep consultant who won't just try to cry it out they might have a new perspective. We were stumbling around trying and buying different things and so paying someone was mostly about getting oversight in our approach. Good luck though. We will all get through it one way or another.