My nephew and brother-in-law left Yosemite Valley(4000') 6/21 en route to Whitney. They hit Toulumne Meadows (8500') @ mile 23 and pulled out due to high snow levels (to the bottom of their packs). Their highest elevation was only about 9000' @ Cathedral Pass. You will be starting 3 weeks later than them and be 15 miles SE. Your intended elevations are Shadow L.8737', Ediza L. 9265', Thousand Isl.L. 9830', Garnet L.9678'. You could continue north up the JMT over 10170' Island Pass (1 mile NW of Thousand Isl. L.) into the lovely Waugh and Gem Lake Basins, but they are similar elevations. If you only go as far as Shadow Lake, as you suggested, check with the Ranger Station to make sure camping is allowed there. Unlike the good ol' 70's when I made many trips over the Minaret Lake Loop, I think camping is prohibited at Ediza and Shadow to allow vegetation to recover from over use.
If it were my trip,I would weigh lightly the expense of permits and cannister reservations and weigh heavily the expense and sacrifices made by your party to make yourselves available for this trip. The Sierra is full of lower elevation adventures and even 10,000' trail heads on the East Side, along Hwy 395, (such as the Little Lakes Valley trail head) are heavily melted out. Talk to your Ranger Station shortly before your hike. My nephew's team is burning up valuable wilderness time trying to find a suitable trail, getting back to their cars etc. because they didn't modify their plan in response to the unusual snow load this year. I'd hate to see that happen to you and your team.