She's exhibiting an upright posture and slightly extending her neck, beak open, to breathe. It's rhythmic, not a one-off movement.
My vet got back to me and he wasn't able to pin down the exact cause. To rule out gapeworms, he gave her a dose of ivermectin, and to rule out a bacterial infection he's started her on Baytril. He mentioned that she hadn't been gasping since he gave her the Baytril (I'm taking that to be a good sign). Her symptoms line up perfectly with those of mycoplasma gallisepticum: M gallisepticum has a very typical appearance with upper respiratory symptoms, a clear nasal discharge, occasional sneezing and, very commonly, sinusitis
One other thing that has us concerned is during a fecal exam to check for parasites/bacteria, he saw a distinctly protozoan object under the scope. He couldn't identify it, but I'm hoping that it's not something fatal like cryptosporidiosis. He's going to send it off to the lab for a definitive diagnosis.
I know exactly what you mean about the "going forward movement", because mine will do that when they've scarfed down a too-large piece of their crumble and have to force it down their throats. I have a BQ hen that is a total pig and is constantly making herself do that lol. I don't know if sprinkling some probiotics on their food would help? It certainly would not hurt them.