I have problems with hand clapping too. Not all clapping, but just a certain kind. I don't really know how to explain it, but it's usually one or two people close to me who clap in a way that hurts, not all clapping in general.
This has recently become a pretty significant problem because my sister started color guard and I'm expected to attend her football games on a pretty regular basis. For last Friday's I got my dad to bring me a pair of earplugs from his work. It was amazing what a difference it made. I could actually relax. I didn't even know how tense and stressed I was until I wasn't anymore.
IDK about using headphones. I guess it depends on the model of yours and how well they block out noise, but earplugs aren't too hard to get a hold of, they're less noticeable, and they'll probably work better. I think it would probably be easier to get your professor to let you wear earplugs than headphones. XD
How about asking people to do a sign-language hand clap for you? (Where instead of clapping, they put their hands up in the air and shake them.) I think it would probably be easier to get people to do something as an alternative to clapping as opposed to not clapping at all, just because we're so conditioned by society to show appreciation when something like that concludes.