Easy as there is only Hwy 287/89/191. From Coulter Bay to Burley is also limited to Hwy 22 to Hwy 26 and I-15 to I-86. Only real option to avoid Hwy 22 out of Jackson is to go back to West Yellowstone and then head south on Hwy 20.
The Tejon Pass on Hwy 22 west does have 10% grades for several miles on each side of the pass. Roughly 3 miles past the pass the grades are in the 6-7% range and continue for another 4 miles. It may be nerve wracking but it is doable if you keep your speed down using the engine braking as much as possible and keep your speed at 25 MPH for the 8 mile section which will take "only" 20 minutes.