Do not take Teton Pass. It's extremely steep and not worth the chance. We did it once towing a 17' TT and were scared out of our wits and that was before we knew any better. We travel from the Boise area frequently to Jackson and beyond and we always take 26 to Alpine Junction and take 89/191 into Jackson. It's a much better ride.
Yes, driving through the park is not a problem. The roads are good and any of the elevation changes going over the Continental Divide feel minimal. You can always do the reverse by taking Hwy 20 into West Yellowstone, go through the park, then down to Jackson. You will climb Ashton Grade but it's doable. Looks scary when approaching it, but it's not really that bad - way better than Teton Pass.