I think you should just use your home. Babies get overstimulated really easily, and it's best to be close to home for them to lay down or have a place to relax. Going out even for an hour could bother them, then you'd have a cranky kid on your plate.
Well, since you're not having food it wouldn't work to go somewhere like the Airport Inn. I'd suggest asking one of the churches for a space to use. They often don't charge a thing, but may ask for a cleaning deposit which you'd get back after you clean it up.
check with the library, too. They might have a meeting room you can use. (The one on Glendive does--they charge a cleaning deposit, but if you clean up and put everything back where it was, you get your deposit back.)
What about the back room at the Montana? If you make this an annual tradition and save all of the empty bottles and cans when your son is 21 he will have enough material to build a house or maybe part of a flood protection structure.