Wow! Your child sounds just like mine. My dd is 3.5 and from the time she was 1 month old she has had some problems. She was diagnosed with Apnea at 1 mo. At 4mo she was labeled with failure to thrive. At just under a year she was diagnosed with BOTH bradycardia and tachycardia. All along the way we have been told she will out grow it...she never has. She had her adnoids and tonsils removed at just over 2.5yrs, her apnea is better, but still there. Her weight...8lb1oz at birth, 1 year she was 12lbs, 2 yrs she was 14lbs, good news today she weighs a big 31lbs!

. She has been through sooooo much testing, everything from bloodwork to ekgs, eegs, etc. She has been tested for Cystic Fibrosis, Celiacs, Dwarfism, heart disease, seizures, etc. We were pretty sure as parents it was celiacs...but tested showed negative. Found out, it is likely it wont show positive for celiacs until they are older, usually 3-5 yrs old, and only if you havent changed their diet by then. To help our daughter with her weight we tried all the nutritional supplements and she liked Kid Essential the best, she would drink 2-3 a day plus Ide put as much food into her as I could. She no longer needs the drinks, although she likes them and we buy them every once in awhile.
Sorry to keep rambling. My advice....ask about nutritonal supplements, they are expensive, some Ins will cover, ours didnt.

Talk to Dr about what steps to take next and do not count on knowing exactly what the answer is anytime soon. Keep input and output logs (what and when he eats and drinks and what and when it comes out). Keep pictures or videos of anything you think may help. Keep on the drs and go with your gut. If you are uncomfortable with what they want to do or say, find another dr. You have to be your own childs advocate.
Good luck