From your description, it sounds like the Uintas. If instead Zion or other southern Utah locations, it would be entirely different.

In many years there is a brief wintery storm that comes in any time from mid-August to mid-September that may drop some snow that usually melts off in a few days. Just be prepared. Wind can be a big issue. Not that you need an all-out mountaineering tent, but smaller is usually better and a full coverage rainfly is good. A wind shirt is also nice to have and they weigh nearly nothing. Because of the wind, I do not use ponchos. I am a real fan of gaiters - I wear knee-high gaiters in all weather - even in the Sierra. Low cut hiking shoes are OK if you have gaiters. Even if it only snows a bit, the wind makes drifts that you may have to walk through. These storms often start with a day of steady rain that turns to snow. Try not to get soaked because when the temperature drops this is a perfect condition for hypothermia. If need be, just sit out the worst of the storm. But, you may have days upon days of perfect weather too. Never can tell. At least get the latest long-term weather report before you start.

The summer convective storms are usually done by late August and early September. But, it is still a good idea to be over passes before late afternoon. I suspect you will get some nights below freezing. I would not go with anything less than a 20F degree bag.