Some great replies here, giving me some ideas.
To clarify: he is fine with being ridden. No soreness, no hollowing of the back. A saddle fitter evaluated him and the saddle and says the saddle fits him well. I got a Thinline saddle pad to improve shock absorption. He really is a lovely horse to ride and actually seems very happy being ridden. His disposition changes completely once I get on him - like he forgets all about the girth. We can even mount him easily and he doesn't move an inch.
Also, the vet (and the trimmer, and the equine massage therapist) looked him over and did not feel there were any signs of ulcers. They did not scope him because in their professional opinions, it is highly unlikely. Sure, I could push for some expensive tests, but I am coming off as an overprotective horse-mom already and I have to trust in their professional opinions or else I'll be alienating all the horse experts around me (he was actually seen by two different vets who said the same thing). We have added a probiotic to his diet because he has loose stool. Maybe that will help?
As for the behavior itself, it isn't severe, as in, he doesn't try to turn and bite me when I'm girthing him. He just does it whenever I'm near his head, like when I'm leading him to the indoor. And it's not like a teeth bared, ears back kind of movement. More like a little head jerk and touching with the lips. After he's ridden, I can lead him around all I want. I can also do it with his halter on - never any attempt to nip them. At first I thought maybe it was the bridle, because he would try to avoid that, but I worked on it and he is now happily putting his head down into the bridle and basically putting it on himself so clearly that doesn't bother him.
Also, I've tried the elbow, but he knows I'm going to react when he attempts a "fake-nip" and pulls away before I can reach him.
So from reading all the replies, I'm going to take it slow with the girth. I may have tightened it too quickly when I first started doing it because it had been so many years since I'd had horses. However, he was girthy from the day we went to see him at the seller's - she tacked him up in front of us and I could see him react when she tightened it. So this is not a new behavior. He just doesn't like having that girth tightened. At this point, I'm just chalking it up to sensitivity.
The moving around in the arena remains problematic. I do put my arm through the reins and pull on the right rein as soon as he shifts when I'm pulling the girth up from the left, but it doesn't keep him from taking several steps away from me. I may try the treats so I'm rewarding good behavior rather than punishing bad. What kinds of treats can I give a bridled horse? Can he have a piece of carrot or will it be awkward? Because he is definitely food driven! I would also consider clicker training as it's something I've done with my dogs, but a verbal cue seems easier because I don't have to use my hands. He's very smart so it will not take him long to figure it out. Maybe if he associates the girth with getting treats he will be more accepting?