If you don’t have the option of flexible start dates, then consider reserving a permit for an alternative trailhead. I can think of two options that might be available if the Happy Isles slots are already taken:
Glacier Point -> Little Yosemite Valley - You hike down along the Panorama Trail joining the JMT at Clark Point between Vernal and Nevada Falls (you aren’t missing much).
Cathedral Lakes - This would require starting at Tuolumne Meadows and hiking to Yosemite Valley, then hitchhiking or taking the shuttle bus back to Tuolomne Meadows.
This would at least allow you to start on a fixed day (helpful when planning airline reservations, etc.). If you can get to Yosemite the day before, you can always see about changing your permit to Happy Isles.
Snowball