Dude,
Everybody's body responds a little, or in some cases a lot different than others. In my experience my performance, i.e. strength and stamina at altitude, increases each day on multi-day outings. In other words, biking White Mountain could be a good acclimitization exercise for hiking Whitney. I biked the 7 mile stretch to White Mountain from the locked gate a few years ago. I had to walk and carry my bike about 50% of the way up. But I did ride 100% of the way down! Some of the riders in my group were able to ride more of the terrain on the way up. On the average for our group I think it took us about 5 hours to go up and about 1 hour to ride down. Your plan sounds like a fun trip, good luck! -Rick