Updating this post:
I've installed the GB client and have been able to change the region in the options to "The Americas", so I'm pretty sure I'll be able to play on US servers with the GB client.
My only concern now is whether using the GB client will cause me any other problems. Will it affect my "home" region, or make me locked into the EU RMAH?
Also, if I start playing on the GB client then download the US client in say a weeks time, I'll still have all the same characters, right?