@boondocker97 pretty well covered it. My favorite place to ride is Top of the World - closer to Red Lodge than Cooke, but about the same difference because I usually take the route mentioned (head towards Cody, then take 296 to 212), which drops you off about halfway between the two. I have rode up from the Red Lodge side, but only once. It wasn't difficult the time I went, but I hear it can be very treacherous sometimes, and that's before you factor in getting caught in weather. I've heard there's a guy in Red Lodge who goes that way more frequently, I think he owns a mechanic shop there, but I don't know him personally.
I guess the quick answer is, the short way is a possibility, but don't plan on it. I think there are some places you can ride closer by, but the trouble is that you tend to run into either wilderness areas, or not have enough snow most years. The good news is, two hours of driving from Red Lodge gets you access Cooke, Top of the World, or the Big Horns. Cooke has the best reliable snow and longest season in the area, and all are great places to ride when the snow is good. Of course, if you're picking purely on access for snowmobiling, Cody would put you closer to everything. Some other places in MT might also be a better fit depending on the job aspect, but unless a couple hours of driving is too much, Red Lodge is a decent place to be.