If you can, you should C2C Forbidden peak. You'll get plenty of scenery and you'll avoid having overnight gear which would otherwise diminish the awesomeness that is Boston Basin.
If you were thinking TFT, I can see worthiness of a bivy up there on route or as a camp-to-camp.
For Forbidden, you should be fine in late-May (pretty early season) or June. Late-June is probably when that gulley starts to go out, but it certainly gets climbed well into Sept/Oct with shenanigans. Unless fresh/recent snow, route is probably snow free (enough) pretty early.
Prime skiing up there is in Apr-May - thus best snow conditions for the approach.
No real alternate sites, though you could do Quien Sabe on a different day (before/after) with a camp in the alpine - that gives you another area to consider for the permit.
Permits are obviously hit or miss, though mid-week you're probably fine. It's busy, but not THAT busy, especially mid-week.
If you're flexible and strike out with the permit, you can always C2C, plus you have Eldorado and other incredible climbs/hikes all over that area, giving you another chance to try for a permit a day or two later.
The area is magical, good luck.