And make sure not to tell them why you're getting the gym membership. They tend to frown upon people using their facilities as their only shower.
Why? This should be ideal for them. Gym memberships are all about over-selling access to assets. If you're not going to use the machines but they collect the same monthly fees, they should be excited.
I read somewhere that fitness gyms are about cycling people through their facilities as fast as possible since you could never fit all the people that have memberships there at once. If anything I'd think they'd love to have a guy who pays full membership go there just for a shower.