Bear canisters are required on both trails.
Shuttle is the big issue with the High Sierra Trail. It is roughly 300 miles of driving between Sequoia and Whitney Portal trailheads. There are a few commercial shuttles that could help you out but the cost would be terrible. Search the Whitney Portal Store's forums for more info.
I haven't done the HST, but have hiked from Kings Canyon to Whitney. The drive back took a very long day. If it were me I'd do a loop that starts and ends on the same side of the Sierra. One possibility is HST to Whitney then up the JMT to Vidette Meadow and down Bubbs Creek to Kings Canyon. That gets you the whole HST plus a nice piece of the JMT. It only adds about two (possibly three) days, and one of those would have been spent driving back.
This presumes you have a car waiting. If you fly to the Bay Area, then Amtrak to Visalia, then take the Sequoia Shuttle to Giant Forest you wouldn't have to get back to the west side at all. You could grab the M-W-F Inyo Transit bus from Lone Pine to Mammoth, then the Yosemite shuttle to Yosemite (spending a day in Yosemite). Then the bus back to Amtrak at Merced.
Personally if I were flying in from far away I would probably want to include Whitney, so that would constrain which piece of the JMT. I would not start from Whitney Portal because hitting 14,495 in the first ten miles of hiking makes me sicker than sick.