That rules out Deso then since you would have a specific launch date, assuming you get a permit. Same with SJ. Hard to say where the water will be in July, usually coming down but like this year was huge. Bugs are still bad on Deso in July.

some of the Idaho runs like SF snake are self permit. same with Ruby-HT and another thought is the upper colorado/pumphouse. Honestly, for what you are talking about with a self-permit and a taste of canyon country Labrynith/Stillwater fits the bill. won't have the rapids but most of the great trips need hard to get permits. maybe do some short day trips on the Arkansas or CO without the little ones to fill that need. Of course John Day was mentioned too. Another thought is the Moab Daily section.