That's like waterproof underwear. It certainly won't keep you dry, so what good is it?
Not that I've ever found a good use for a baseball cap in the first place. Maybe if your eyes are unusually sensitive to sunlight... but aren't you in the UK? Should never be an issue for you.
A normal visor is sometimes nice if you think it will be raining and you REALLY want to get some rock climbing in. Been there done that at Index, Little Si, etc. Will not interfere with a helmet. baseball caps tend to do so IMO. Obviously only works on established routes and any alpine adventure will see falling stone and its a no go or its a get the Hell out of there and the last thing you are thinking about is getting a little wet of which a baseball cap is not helping anyways.
Waterproof hats are worse than normal hats. Hotter and hold more sweat. There is a reason that old timers had BIG brimmed hats that went outwards on ALL sides. If you want a hat to wear when it is raining look no further than the large brimmed hats worn by peoples in the tropical regions. You can buy them online. Never find them in a store.