I have not driven the route you suggest, nor plan to. I would just drive thru the park. The reason why....
This from Mountain Directory.
"12. TETON PASS elev. 8429'
(on Wyoming Highway 22 west of Jackson, WY)
Teton Pass has sustained 10 percent grades on both sides of the summit. An unusual feature is that both runaway truck ramps on the east side of the pass can be used only if the runaway truck crosses the oncoming lane of traffic. There is a posted weight limit of 60,000 lbs. on this pass.
The westbound descent from the summit of Teton Pass begins with a 25 mph speed limit and a truck warning sign--"Steep grade--10 percent next 3 miles--use lower gear." This grade warning is repeated a mile later. About 2 1/2 miles down from the summit the grade eases to 6-7 percent and the speed limit increases. This grade continues for about 3-4 miles and eases near the Idaho state line.
The eastbound descent from the summit of Teton Pass starts with a truck warning sign--"Steep grade--10 percent next 5 1/2 miles--use lower gear." There are 20 mph curves near the top. About 1/3 mile down from the summit is a sign--"Runaway truck ramp--2 1/2 miles LEFT side." This warning is repeated several times as you approach the ramp. To use the ramp you must cross the oncoming lane of traffic. The ramp slopes uphill.
The second runaway truck ramp is about 1 mile after the first. It, too, is on the left side and is an upsloping ramp. At this point the grade begins to ease to about 6-7 percent and the speed limit increases. The hill continues to the town of Wilson, which is about 5 1/2 miles from the summit."