I'll second the Northern Loop suggestion (Rainier NP). I've hiked it 3 or 4 times--never had problems getting a permit without reservation. Going mid-week helps. Part of the loop is on the WT, part is in a more remote corner of the park. In my view, the hike across Windy Gap (not part of the WT) is one of the best in the park.
Solitude won't be perfect, but I wouldn't call the Northern Loop crowded, at least not relative to some sections of the AT or (I presume) the more popular CA trails. Depending on day of the week, time of year, and portion of the trail, you might see only one or two other people per day while hiking the Northern Loop.
Sides trips could include hiking out into Grand Park and to the natural bridge near Windy Gap. Closeup views of glaciers (and of course, Mt. Rainier) are also a plus.
For a longer trip, you could make a figure-8 around Mother Mountain and the Northern Loop. My trip report on that route from last year is here:
Note last year was a big snow year--conditions this year should be somewhat different.
One of the nice things about Rainier, given your requirements, is the opportunity to get above timberline and experience alpine hiking without extreme elevation--less than 8K at the highest points for these hikes. Much of the Northern Loop is in Mt. Rainier's rainshadow, where weather *tends* to be drier and (in my experience) a bit more predictable. But Mary D. is right--the mountain makes its own weather, and anything can happen. For summer, highs will rarely break the low 70s but alpine/sub-alpine sun is intense. Rain might bring temps down to the low 40s (hypothermia risk). The designated campsites are all below treeline, but nights can get pretty cold. I've experienced light snow in mid-September.
One potential drawback: the Northern Loop is best accessed from Sunrise (east side), a bit of a drive from the Seattle area relative to Paradise.