I am not anti- tom thumb. I am one of the few people on here that doesn't think it is evil.
And I don't see any harm in trying it since your trainer thinks it might be good to try and he/she knows the horse better than we do. However,
in this case if the horse is not happy in a snaffle he will probably not be happy in the tom thumb because they have the same mouthpiece.
If you have other bits to try, I would try something either with a 3 piece mouthpiece OR a more solid mouthpiece. If your horse can take a curb bit (which a tom thumb is), I would try just a very simple plain grazing bit with a solid ported mouth.
Or go the opposite direction and get something with a 3 piece dogbone or lozenge mouthpiece. If someone has these bits you can try, I would try them all to see what he fusses with less.
It could be that the nut-cracker action of both the snaffle and tom thumb won't work well with him so I would try a different mouthpiece personally.
But there is no harm (that I have ever found anyway) in trying different bits. I do it all the time. Just for fun.
So don't be afraid to try something with a different mouthpiece. I have never found it to do any harm and you just might find a bit your horse is at perfect peace with.
PS. The neck stretching/pulling thing to me says he's not happy in the bit and he is trying to shake off your contact. So either you have too much contact without relief all the time or he hates the feet of the bit. Or it's become a bad habit because he knows if he pulls he will get relief. Either way I would try to find something he doesn't hate the feel of. And try giving him more loose rein and just work in contact here and there, not the entire time you ride him. Get him giving to contact in small amounts of time and then build on it, with loose rein as a reward in between times if you can.
He sounds like a horse that is defensive about his mouth from his past riders.