Here's another option: There are some Mom & Pop campgrounds in Livingston that would have showers- And you can get from Livingston to Bozeman about as fast as getting from Hyalite down into Bozeman, and a whole lot safer. When driving to Hyalite you have to watch for half drunk college kids and local rednecks racing up to the reservoir, and then tearing back down when they are totally drunk.
Livingston isn't a bad town and is within a 30 minute drive of actual and real Montana. You can buy a 3 day license and catch some nice trout between Livingston and Gardiner, you can see the Crazy Mountains and there are some good places to ear there.