As a bike rider and one who has stayed at Upper Teton Depends on your bike, and your skill.
The last section is a dirt road and can be rocky in parts, but it is not hard, nor a long section of road. When you first turn off the highway it is a straight packed gravel road until the Y turnoff. (stay left) and then you weave upwards through the trees. It is this part that is more loose rocks, but its not sandy or soft. Its hard and firm with a few bumps and dips.
We did it fine on Goldwings. But the view is so worthwhile if you are lucky to score the spot in the middle up against the ridge. If you have an adventure bike, no issues at all and you will have a blast.