We did the High Sierra Trail last year. We used the shuttle service from the Wilder House to get from Whitney Portal to Giant Forest to start the hike. We had to drive through Mojave to get over to the western side, so if you can get there I'm sure they could do it. I don't know if they pick up in LA, but it's worth a call to see. They were priced reasonably and on time.
http://www.wilderhouse.com/If you go in from the West, I don't think acclimating is that important, I think it starts at about 6700 ft. We did the HST at the end of Aug in 6 days, but you will have snow at the end of May so it will probably take a little longer.
I was trying to post a pic of the view of Whitney from the west, but don't know how. Any hints from anyone on how to do this?