Welcome, human!
That's quite a big loop you plan! It will give you a full complement of the Sierra experience.
The permit you have already is all you need. You could even exit the Whitney Trail with it for a resupply (though it's a bear climbing back up to the crest with a full pack). That Trail Crest Exit Permit is only required by Inyo rangers if you start on the east side on any Inyo N.F. trail head.
The crest of Whitney where you would be hiking is all in Sequoia National park, anyway, and is administered by the Sequoia rangers, not Inyo.
