Thanks a lot Hobbes and Steve!
I did try calling White Mountain two or three times this week and each time no one answered with no opportunity to leave a VM. I'm going to keep watching recreation.gov to try to get the perfect permit. I'm actually pretty flexible with any 10-day window from August to mid/late September. If that doesn't work, I'll just plan my trip for the dates I have, assuming I'll have 10 days, and get to the station at open the day before my entry (confirmed at South Lake at this point), with the hope that they can modify my exit date at WP.
I'm also going to look up the Mountineering route. I saw it on Caltopo but I don't know much about it otherwise. Any good resources you can recommend?