So many options... As Will said, I would avoid anything along Mirror Lake Highway if avoiding crowds is high on your lists. Mosquitoes are non-existent at that time of year. Even in a late snow year, they're pretty much gone by early August.
If we're combining fishing with avoiding crowds, I have two suggestions, both are entrances along the south. The first is to use the Swiftcreek Trailhead and hike up to Timothy and/or Farmers Lakes. The fishing is great up there and those lakes rarely see much in the way of traffic. There are about a dozen lakes in that area that are worth fishing and making a basecamp in that area for a few days would make for a nice trip. It is about a 12-mile hike in, there are a ton of lakes to explore, and there is an option to make it a minor loop.
The second suggestion would be to use the Lake Fork Trailhead and head towards Brown Duck Basin and potentially Squaw Basin from there. The TH itself can be busy because of Moon Lake, but almost no one hikes into the Uintas from there. The fishing is fast at Kidney Lake (there are a few kidney lakes in the Uintas). On your way out, stop at the north end of Moon Lake and drop your line in there as well. Good luck!!
