Driving past Grand Targhee means you're going up and over Teton Pass on your way out of town. It's a twisty mountain road with a lot of elevation change, but it's also a high-traffic route and (I assume(?)) maintained accordingly. I've only driven it once, en route from Jackson to Grand Targhee (and then home to Red Lodge) last Christmas, but I did get to wait an hour or so for avy control work. The drive from Grand Targhee through West Yellowstone and to Big Sky got a bit interesting at times given that we hit weather, impacting both visibility and road conditions. I was feeling like a pretty competent winter driver doing 45-50 MPH on wall-to-wall packed snow until I got caught by a guy in Super Duty who disappeared in all of 25 seconds after I pulled over to let him by, so I'd expect that you may find a rental with crap tires to be interesting in certain weather conditions. Jackson -> Big Sky isn't a bad drive, though; Jackson -> ski at Grand Targhee -> drive through Big Sky and continue to Red Lodge made for a long day.
Be aware that the road from Big Sky to Bozeman is curvy and subject to slippery patches; if you're trying to make a flight, make sure you leave wiggle room both in case you encounter slippery conditions and in case someone else screws up and blocks the roadway.
The other winter route from Jackson to Montana (Togwotee Pass, through the Wind River Range, and then north) makes zero sense if headed to Bozeman.