If money is a concern for housing costs, getting more than an hour in any direction out of Billings is a good idea.
Bird hunting is best in central to eastern Montana, all things considered. there are some isolated upland spots in western Montana, outside of mountain grouse hunting. access can be tougher on the west side , unless you have access onto private land.
Personally if I wanted to bird hunt, and have a great place to live, the Lewistown area is hard to beat, app. 100 miles north of billings, a small regional airport, hospital, a couple good restaurants! And great people, wished I lived there. Housing isn't driven up , as in cities with more jobs... Great falls would be second choice, larger city , and all the good and bad that goes with 70,000. folks