Travel Note
Both the Bozeman and Billings airports are around an hour drive away from Big Timber. You might have better luck finding more affordable rental cars in Billings. Although there aren't a lot of options to stay in Big Timber you can always book an Airbnb or VRBO. You can also look into staying in or around Livingston, which is around a 45 minute drive away.
Flight
Flight
Hotel
Hotel
Hotel
Hotel
Hotel
Hotel