One thing I did with my son when he was about that age is he took a late nap (like 5:00 p.m. to 7:00 p.m.) and then he was up for a few hours. I gave him a feeding and then we both went to bed at the same time, around 10:00 p.m. That way, he was only having one feeding that fell in the middle of my sleep.
I would also say try chiropractic care--for both his acid reflux and his sleep issues. Have you tried the "sleep slings" that go inside the crib for acid reflux? Whatever route you take, I think you could benefit from a solid night or two of sleep ASAP. Do you have a friend or relative who could take your baby for a night, or maybe you could swap houses and she could come to yours and watch your baby?
One thing I wonder about is what everyone's (you, your doctor, everyone who responded) definition of "through the night" is. How many hours are we talking about?
You said your book says something about 95% of babies sleeping through the night at a certain age. Well, someone has to be that 5%, unfortunately. I'm not saying it's YOU, but statistically someone has to be.
Lastly, one risk you take with him crying it out during the night is then he could be awake for quite awhile, and may compensate during the day by taking longer or more frequent naps. One thing to think about is how long does it take to feed him and have him return to sleep vs. letting him cry? I'm not against crying it out, but to be honest, if he eats and then falls back asleep, I would just feed him. If he is up for awhile and it gets to be a fiasco, that's a different story.
Good luck.