I spent multiple nights there last season in a camper van. Of the 20 hills I went to on that road trip Big Sky was the worst for the "No overnight" parking rule.
That being said, if you look on google maps there is 3 completely empty cul-de-sacs side by side with zero traffic or houses.
Road names are Speaking eagle,  spotted eagle and black moon road.
I slept there 3 nights in a row with no problems. Very quiet and peaceful!
However if you're like me when I was doing the trip, I much preferred sleeping parked next to others that were camping, made me feel more safe. If you drive Lone mountain trail, which is the road between the ski hill and the town, there are multiple pull off spots along the way, it wasn't hard to find a spot where another camper had pulled off for the night.
I would recommend doing what I did, when your ready to go park for the night drive Lone mountain and see if you can see someone else parked for the night, and if you see nobody go to the quiet cul-de-sac.
Hope this helps!
Enjoy Big Sky! its still to this day my favorite of 50 hills i have been to.