Thanks for the feedback and info, everyone. It looks like White Mountain is a no-go for this challenge unless I bike to within a mile or two of the summit (and even then, my chances look iffy).
So that leaves Langley. New Army will be melted out by early July this year, but Old Army might not. There'll be 14 hours and 20 minutes of daylight to work with at that time. Subtract 20 minutes for traffic and finding parking, subtract 3 hours of driving each way, and subtract another half hour for miscellaneous things like unloading gear, filtering water, and breaks for food and summit photos.
That leaves 7.5 hours of actual moving time on foot. It probably wouldn't be too hard to jog the part from the trailhead to the base of New Army or Old Army, but everything after that needs to be hiked at a steady pace unless you have superhuman levels of fitness and coordination.
Rough gameplan -
First 6 miles, TH @10,100' to base of NAP @ 11,100' - 1.5 hours, 15 min/mile pace
Next 5 miles to summit @ 14,026' - 2.5 hours, 30 min/mile pace
Next 5 miles back to the base of NAP - 2 hours, 24 min/mile pace
Last 6 miles back to TH - 1.5 hours, 15 min/mile pace
Total - 7.5 hours
Seems doable, but it certainly ain't easy!
