This is maybe not the answer you're looking for, but it is based on my own experience, might give you some reassurance and will hopefully spur someone on to write a better answer.
Backstory: My child was a "happy chucker". Nothing physically wrong but they could vomit quite quickly and would do so when upset. If I'm honest, I probably used it as an excuse not to sleep train at a time when it felt like there was a lot of pressure to get them "sleeping on their own". I did the calculation and figured out that a few nightly minutes sitting with them was less hassle than (possibly) a few nights of mild distress and lots of washing. In hindsight, I might have got that calculation wrong, but I don't regret it.
Instead, I made an attempt at a "gradual retreat" method with a bedtime routine. Started off holding them in the big bed and transferring them to their cot. Then into the cot with me right next to them (hand holding), then a little bit further away. I made it as far as a chair in their room with them in their own bed, armed myself with my (muted) tablet and would sit there, after the bedtime story, reading until they fell asleep - about 15 minutes of "me-time" nightly. It wasn't a hardship for me (introvert!), and other-parent was capable of following the same routine. Sometimes we'd even offer (or compete!) to do bedtime.
Rules were no talking because "it is sleep time now" (I was strict on that), no getting up (unless for toilet or emergency), and I used some sort of lullaby music, too. I was also very boring during that time - I was there, but I could have been asleep for all I was doing.
I'll warn you, though, doing it this way, you're looking at months or maybe years of boring nightly me-time with your tablet. But my offspring (eventually) went to bed at bed-time and stayed there until "sun up" - even on Christmas Eve/Day!
I didn't solve the problem quickly, but I did solve it gently. Just like a lot of things, one night you'll put them to bed, and it'll be the last night that they need you to do that. You won't know which night that is.
Some of my ideas probably came from "The No-Cry Sleep Solution".