I would be in for a cap to, but maybe a bit more understated compared to the tops.

If it means anything my thoughts would be something along the lines of that one you posted already. Maybe black with a green stripe/s and the logo smallish on the side.

Not a fan of cycling caps, but I'd go a long sleeve jersey as well if there was one.

Been meaning to buy a cap for ages, a bb one seems a better idea than most!