You've already started. Yeah, that Mono Meadow trailhead (trailhead #34) looks to be pretty clear for at least a fair part of July. Find the dates that work for you, book it! Call it first night camping for the permit application as Illilouette Creek. Exit Happy Isles. If you think you can do it in 4-5 days, give it 6 days out from your entry date to cover contingencies. It's also a lot more likely that in the outside chance that you don't get a reserved permit for that trailhead that you can get in on a walk-up permit with better odds than most of the other trailheads coming out of or around the Valley or even Tuolumne Meadows.
Another trip to look into is Ten Lakes Basin. Once you check it out, if that sounds better, but the entry at Yosemite Creek is all booked, come in from White Wolf, but it's going to add almost 6 miles more overall.
Look at pictures of both areas online. Get that map. Try to find the Tom Harrison 1:63,360 of
Yosemite High Country. You should be able to get it/order it through the REI you guys have fairly close to you. Here's another link for the really basic general overview of the trailheads/trails in the park, but it's not a substitute for the real thing.
https://www.nps.gov/yose/planyourvisit/ ... eID=292137You guys won't regret either route. If you guys get it decided and booked, you won't have any shortage of suggestions on everything you need to know.