Wayne, you would know the answer to this. I have always assumed that the sun/moon rise/set calculators did not give true times, but times based on the location transferred to mean sea level--directly above or below the point of interest. So true only for the earth being replaced by its geoid.
For example, your calculator predicts sunrise at Lone Pine on July 19 for 5:52 PDT. I assume that the sun actually rises for Lone Pine t1 minutes earlier, because it is at ~4000' instead of MSL, corrected by t2 minutes later because of the Inyos. So actual sunrise is 5:52 - t1 +t2, where t2 would be greater than t1.
Similar for Mt. Whitney, but its t1 is larger than Lone Pine's, and its t2 is very small--nonzero not because of the Inyos but the Spring Mountains perhaps.
True or false?
-----
Jeff, be aware that the moon will be low in the sky (it's summer), so hidden by Mt. Irvine's NE ridge for the first part of your hike. I haven't checked, but I'd guess up to around Trailside Meadow. Of course, it will be brightly illuminating the walls on the north side of the canyon, so will help somewhat.