The Jenny Lake Rangers blog is always a great go-to for conditions in the Teton backcountry, and according to them as of August 9th:
“the trails and passes throughout the park are primarily snow free, and the mountains are filled with hikers and climbers. As of right now, the Teton Crest Trail is essentially dry and most hikers will find little use for an ice axe along the entirety of the trail, including Paintbrush Divide. With that being said, expect to cross some snow and steep, loose, wet rock along the Divide, which may be difficult to navigate. In short, it still remains an area of caution.”
From my friends there and which the blog confirms, the area has seen some increased precip this year beyond the regular storms, with “significant” rain and thunderstorms.
As for public transportation, a bus goes from Star Valley to Jackson that you might be able to hop on if you’re willing to walk/thumb a ride to a nearby stop (probably in Wilson), and then through Jackson proper (your risk tolerance for both hitching and walking down that particular highway will vary from mine, natch!). The park shuttle got bought out in 2016(?) and now seems to be more for tours and for private charters, but it is available as a private operator (link to Alltrans site).
There are a few taxi services, which of course are not public and may be $$$. Jackson Hole Taxi is the one I’m most familiar with, and they’re willing to drive pretty far. It might be worth comparing that price to a) catching the bus to Jackson and then b) getting a rental car in town. There’s also uber/lyft in Jackson now but cell conditions near the pass may be questionable. Another option is to reach out on Craigslist and see if there’s a local willing to make a buck dropping you off at Flagg Ranch so you can leave your car at your endpoint instead. You may have to get creative and mix’n’match options up a little.
While personally I’ve never had the best of luck hitching in that area, if you’re considering that as an option you might get lucky with a local via the Moose-Wilson road into the park and then from there to Flagg Ranch.
I’m going to be in the area that week as well, although I’ll be starting up Grand the morning you leave so alas unable to help you that particular day. But if you end up truly desperate DM me before Aug. 22 and mayyyyybe we can figure something out?