I have a ton of advice for you. :-)
My daughter, E, is a micro-preemie (born at 1#, 5 mos early), so she has had 2 barium swallow studies so far, and we are going for our third here in March. They are NO BIG DEAL. So don't worry over them. It will be okay. The worse thing about them is they want her hungry, so she was, very! So a little fussy. But it shouldn't hurt him, or bother him.
E was aspirating since birth on everything, so we thickend her breastmilk with thickner. Even then, she had the vomiting issue like your son. We have used "Simply Thick" gel thickner in the past and are now using "HydraAid" gel. They are both great, but the packets are easier to empty with HydraAid. E is on Medicaid so we do not pay for it, but your insurance may cover it, if the swallow study deems it Rx. If not, it is about $30 a jar and that will last you almost all month, longer if you are on 'nectar-like' thickness or thinner.
At her 2nd study, done in Oct, when she was 8 mos old, she was STILL aspirating on a 'nectar-like' thickness, so we have been bumped up to 'thin-honey' since then. She has that same "lazy-larynx" issue, also refered to as a "soft larynx" (FYI) and they told us it only gets worse as they get older (which seems counterintuitive) until they are big kids. Aspiration is very serious and can cause fatalities from Pnemonia and other chest infections. So do not take it lightly. The technician at E's Oct study was very upset that she had still been aspirating all that time, and we were very lucky she had never gotten sick.
I know what you mean about the constipation, so I would thicken with a thickner rather than formula if you can afford to, it's almost the same price actually. The thickners will not constipate him. But it will slightly dilute your BMilk, so I would add a little bit of formula to off-set that. We are no longer allowed to give E Bmilk due to a protein allergy, so now we do a ratio of 20oz H2O, 15 scoops of Elecare Rx formula, and 22 pumps of HydrAid for a 'thin-honey' volume that gives E 22 calories per oz. You would do roughly 4 scoops of formula to 20oz BMilk for a 22-24 calorie meal. Depending on your formula's caloric density.
E poops with every feed on that. So it seems that maybe something that you are eating may be constipating him? Be sure you get enough fiber too, a Benefiber supplement may do you both some advantage.
E was on an antacid from birth to December and I agree with you, I hate to give meds if I can avoid it. It did its' duty though, and you want to protect the esophageal lining from being eaten away by his vomit. I know Bulemics have lifeling issues, so there may be permanent damage by stomach acid going the wrong way as much as he is.
Lastly, DO NOT USE "Thick-It". It adds some caloric value, but gets thicker and thicker the longer it sits and is a big pain if you mix batches in advance. It will not work! :-)
E also had an EGD, (camera down the throat all the way to the small intestine) in Oct and despite her vomiting, it was normal thank God. But she did not vomit as much as you say your son does. She also had a PH Probe that came back negative, so again, I asked myself, "Why the Antacid." But I continued it through Dec anyways to be safe.
Hope this helps, write me back if you have any other Qs. And good luck!!