Hi there. So sorry to hear of your lo's tummy issues and how little rest you're getting. I've been there and offer you the biggest of {{{hugs}}}.
From my experience, and from following bw, sleep training simply cannot occur if your lo is in discomfort. There are many Dr's out there who don't think there's anything wrong with CIO or CC but I can tell you from experience that it was a disaster for us. I'm not telling you this to scare you, but I didn't get BW until my twins were 6 months old and by that point I was desperate. I had tried rocking, walking, bouncing, CIO and CC with disasterous results. (CIO and CC are my biggest regrets as parent aside from not being advocated by BW).
They had undiagnosed reflux at the time and actually weren't medicated for it until they were 14 months old. Only then did they start sleeping through the night. We have gone through so many trust issues since then, and it took a full year of regaining their trust to have a decent bedtime and that just started happening on a regular basis last month. They're almost 3 now, and they were medicated up to 20 months. So yes, most do outgrow reflux to some extent, but it takes a long time.
To be entirely honest, I think your lo needs to be seen by a Dr that will listen to you and your concerns. You have a gut feeling and you are most likely right that the tummy issue is preventing sleep. You are advocating for this lo and your Dr should respect that and your lo in particular. I let my Dr make me feel like I was simply a neurotic mother overwhelmed with having twins (which I was a bit), but there was something else going on that needed to be addressed. It was actually dh who pushed it with my Dr and had the issue taken seriously.
Dr's can only go by what we tell them. This really does sound like reflux to me. My third lo hardly puked during the day but nighttime was a different story. Not all refluxers projectile vomit and lose weight. Some Dr's use that criteria for diagnosing reflux. There's nothing wrong with switching Dr's and nothing wrong with trialing meds.
Biggest of {{{hugs}}} again. I've been there, I've been frustrated and sleep deprived and I can honestly say by getting my 3rd lo diagnosed and medicated properly earlier than the twins has helped so much with her sleep training.