You possibly just accept the 0.05% of your employees who cannot wear headphones are going to work somewhere else / be less performant.

Aside: I still don't quite get what exactly about the headphones is causing you pain? If you have them on, without playing anything, is that painful? (in which case maybe lighter ones that exert no pressure on the skull could help?) Is it the volume? Is there some weird property of sound produced by the headphones that is not present in general noise, that causes the pain?

(please don't take this as me disbelieving you, I'm just curious)