You can get 2 permits, 1-2 weeks apart depending on speed,? from either Florence Lake or VVR.? Use the first one to go SOBO to Mt Whitney - or NOBO to Happy Isles.? Then use the second permit to go the other way.? Transportation is the killer for this technique but the permits are fairly easy to get.
If you're in a group of 3 or more you can set cars at Mt Whitney and Yosemite Valley which would make it very long drives but easy.
-- Byron Nevins Lead Moderator of JMT at groups.io