> Can we access the wilderness via the Whitney Trail, climb Thor Peak, descend to Iceberg Lake and continue to the summit on the Mountaineers Route?

Technically, yes. You could climb Thor from Mirror Lake, and descend to near Lower Boy Scout Lake. But that would be a mighty grunt to haul backpacks up and over.

The North Fork overnight permits has a 10 permit quota, 6 reservable online at recreation.gov, and 4 available for walk-ins at 11 AM the day before the hike entry date. ...and of the 6 reservable, if there are any no-shows, they become available at 11 AM on the day of entry. If I were you, I'd try for an actual North Fork overnight permit.

Now, you didn't say, but it sounds like you might prefer a day hike permit. In that case, like V.Fred wrote, a Main Trail day hike permit is what you need. Since Aug 1 is a Wednesday, you are quite likely to be able to score those on Tuesday (at 2 PM, they make all the no-shows available.) Last year, there were never fewer than 14 unused day hike permits on any Wednesday in August. (Unused Permits link)