If you go on Amazon, you can find a runners cap that will breathe nicely and should be able to find one where the logo is also black so it is almost invisible. (That’s what I did a few years ago.)
I think how a cap is perceived is regional and local. in the US, USSF used to strongly discourage them, I think on the “doesn’t look professional” theory, as the pro refs don’t where them. (That was under the same regime that thought refs having different shirt sleeve lengths was a crime against humanity.) But the pro refs are doing a single game in a stadium. Here in sunny Southern California, if I do more than one game, I almost always wear a cap. Sure, sunscreen does a lot, but with a family history of skin cancer, I‘m protecting my face.