There’s a great loop hike starting from the Twentymile Ranger Station on the south side of the park that checks most of your boxes. It’s just under 19 miles.
Head up Twentymile Creek to the AT (passing Twentymile Cascades at a mile or so which is a nice cascade), make a detour to Shuckstack Firetower (amazing views, but the climb up the several flights of stairs to the lookout cab can be a bit heady for those with a fear of heights), then head over to Gregory Bald. If you time it right and the azaleas are blooming on Gregory Bald you’ll be in for a real treat. Rhododendron and mountain laurel should be showing out along the loop as well. Especially since you’ll pass through lots of different elevations and aspects so you should hit some patches of each that are peaking somewhere along the route. You’d definitely want to camp at Sheep Pen Gap/Campsite 13 near Gregory Bald on your second night so you can hike back up there (quarter-mile or so away) for sunset, stargazing, etc. Campsite 93 would probably be your best place to camp the first night. Not too far and not very steep hike in so you can pack some nice goodies — heavier and tastier food, beer, etc. — without wearing yourself out.
You’d probably see crowds around Gregory Bald (and be sure to book the campsite as soon as it opens up, around 30 days out I think is when the window opens up, it’s a really popular site) and on the AT, but the rest of the loop probably wouldn’t be crowded at all really. Not sure what bugs would be like up there, but I’ve done plenty of trips in the Southeast in June (including just south of the park in the Joyce Kilmer-Slickrock Wilderness) and never found them to be too much a nuisance then, but probably varies considerably depending on your elevation. Definitely bring bug spray though.
Another good area to look at in the park would be the Big Creek area and a loop going by Mt. Sterling (fire tower there, too). I think that loop is around 20 miles or so if you go via Gunter Fork Trail (highly recommend, beautiful rugged trail with some nice waterfalls).
