IgE looks normal so it doesn't look like an allergy (as far as I know).
I'm not sure what the three types of IgA testing were for. I've never seen that done. He does appear to make adequate levels of IgA though so his tTG IgA should be fairly accurate... as accurate as it can be in toddler. His tTG IgA is negative but those tests do tend to miss celiacs who are toddlers more often than adults. The tTG IgA is 75-95% sensitive so there is a 5-25% chance his test is a false negative - probaby closer to the lower end because his IgA levels are adequate. This report discusses it on page 10-12: http://www.worldgast..._long_FINAL.pdf
As you pointed out, he could still have celiac disease in spite of the negative test. Are the other tests available to him? This is the full panel:
tTG IgA and tTG IgG
DGP iGA and DGP IgG (best test for children)
EMA IgA (detects more advanced cases)
AGA IgA and AGA IgG (older and less reliable test thought by some to also show non-celiac gluten intolerance - NCGI)
The DGP test might show you something, or even the IgG based tests...
There is also the chance that he has NCGI in which case he will have all the same horrible symptoms of celiac disease but not the villi damage (the blood tests look for autoantibodies that cause the villi damage). All you can really do is go gluten-free to test that theory- once all other testing is done.
My boys all had negative tTG igA tests, which were the only tests available to us, but I am a celiac and 2/3 of my kids had symptoms so I made them all gluten-free, and they are all much healthier because of it. You may have to go gluten-free without a doctors order; if you do, give the gluten-free diet at least 3 months before you judge it's effectiveness as healing takes time.
Good luck in what ever you decide to do.