The short answer is DO NOT DO IT. The long answer. You can do it, the road is not that bad, but you may need a new set of brakes later -- maybe a new transmission.
A better plan, enter from the East (Cody). But before entering YS, park the trailer at one of the many campgrounds at or near Cody. If you dry camp I think there are a number of places on FS or BLM land. Take a day trip to Red Lodge. The next day take a day trip from Cody to the NE entrance near Cooke City US 212. Do the loop in the park and return by US 14 (the East entrance) to your campground.
You should enjoy Cody, Red Lodge and the drive (without the trailer) Plus without the trailer you should be able to see the wildlife (and the associated traffic jams) in the valleys of YS.
On my last trip, I entered from the West (spent a night at Bakers Hole). Then pull the trailer over to a NFS campground near Cooke City. The drive down the mountain in YS to the NE entrance required me to stop a few time for the trailer brakes to cool off. Later I found that I had wore a flat spot on one set of Trailer tires.
I then visit Red Lodge and Cody without the trailer.
I re-entered the park from Cooke City and left the park with the trailer by the South entrance.
Edit: You can see some photos of the Bear Tooth in my travel photo below. I think YS trip is in the middle.