Try calling the Bishop FS office for info. However, a high altitude hike like you describe just before your Whitney hike isn't really going to help you much to acclimatize. It might merely give you a chance to experience the pain before the higher hike.
I have been in your situation many times while living in the San Francisco area (fast drive to trail-head on Friday night, sleep at trailhead, hike in Saturday morning and hike to camp, climb (!) Whitney on Sunday, drive home Sunday night, go to work Monday morning), with no attempt at acclimatizing. But just did it anyway! How? By going slowly with very early start, and being prepared to descend if feeling sick. It was the price we paid for being "weekend warriors".
Park as close as you can to the Whitney trailhead and sleep there, then suck it up and do the hike, following this advice:
http://inyosar.com/wp-content/uploads/2020/02/PSAR-2020.pdf
