You should just toss that idea out. Hard Pass. Unless it becomes the focal point of your trip.
Mt. Whitney is heavily forested until the rough halfway point - Trail Camp - where the summit comes into view. Meh. However, there is a stench at the scenic alpine lake. Too many humans camping there with corresponding waste. And day hikers stopping. See: trash on Mt. Everest.
The most interesting thing about Whitney is the rocky, above-alpine ridge after switchbacks at 12-14k.
It can be a day hike, but more like a 24-hour day, 4 AM to 4/5 PM. 20 miles/6.7k ft. Otherwise, it's 2 days. And whatever acclimization you want to do - or not. Plus travel to/from Lone Pine, CA, so you need to add Yosemite, Mammoth, and Death Valley to your trip.
Personally, Timberline Lodge skiing and maybe a Rainier/Camp Muir day hike from Paradise might get you what you want easier. Go see Yellowstone afterward.
Jersey Mall is the most reliable option.

Mid September day hike/climb
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Above halfway:
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