The museum in Cody is very popular and probably 2 days to see a lot of it. It's the flattest route via Fishing Bridge. If you go via the Chief Joseph Highway it's prettier but more climbing. Your set up might be too much for the Beartooth although it's worth doing in the jeep.
Bighorn Canyon is lovely as are the mountains which have a lot of moose along 14 between Greybull, Shell Canyon and the Burgess Junction. Probably out of your way, but it's a beautiful drive. Couple of steep sections but not too bad.
If going from Tetons end, Dubois has a great Military Vehicle museum and again you can take 2 days to go around it. That route will take you through Thermopolis a hot springs town and then Cody.
If you end up at Red Lodge over a weekend the nearby town of Bear Creek has pig racing which is fun. It's not overly cheap to eat there but the food is good. You need to book a table.
Hopefully other have suggestions for the area east of Red Lodge but when we drove out that way it was pretty quiet.