Based on the resume you provided, this should really be a walk in the park, unless you are trying to break the FKT for this. The weather in Death Valley is going to be hot really no matter when you do this if you are doing it between May and September. If you are trying to solve for the easiest conditions, I would start the bike ride shortly after sunset (~8pm or so) such that you are arriving at the top of Whitney Portal around sunrise (assuming you are a moderately fast cyclist). This way you will summit around 12pm (give or take depending on fitness) and down off the top in case there are any T-storms.
As for the permits, you can take one of three approaches:
1) Get a permit well in advance and plan accordingly.
2) Be a bit more ad hoc and look to find a permit when you see one available on the site
3) Don't get a permit and risk it (less of a risk if you do the MR)
As for route choice -
Personally my preferred route is up via the MR and down via the MWT. Climbing the MWT after a 8-10+ hour bike ride sounds horrible, and descending the MR after 12+ hours of exertion seems horrible too. Based on my read of the snowfall accumulation this year (
https://cdec.water.ca.gov/reportapp/javareports?name=PLOT_SWC), I would estimate that the MR will be snow free by the end of July (assuming you take the scramble on climber's left on the gully and take the class 3 section on climber's right on final 400).
