We did the Northeast entrance after the Beartooth scenic highway last year and found the less-used approach quite nice. "Scenic" is an understatement for the Beartooth highway.
Overnighted in both Red Lodge and Cooke City due to a late night arrival in Billings. Cooke City is small and rustic, but we had a pleasant stay there.
I wouldn't want to do this route with any time pressure.
In prior years, we've come in through Bozeman happily. Any entrance is fine if you miss the crowded months.