As Steve said, a night on the top can be very special. Plus, you will have a big bonus - the moon will be almost full!
My only concern for an overnight trip like that, especially in July, would be weather. Since afternoon T-storms usually move in from the west side and forecasts are not always spot on accurate, I like to delay the ascent from the Trail camp or Iceberg lake (if I choose the MR) till after ~3 pm and then make a final call when I can see the other side (e.g. at the Trail Crest or the Notch). Of course, it is much easier to commit when the sky is completely clear, but it can be done relatively safely done even when there are T-storms in the area as long as you know where the fronts are moving to and have a plan for an escape. On one night, I had a spectacular view of isolated T-storms over Palisades with absolutely clear sky over the Whitney summit and almost no wind.
A few other things to conceder:
- Although the summit plateau is huge, there are not many flat spots that are big enough for a 2+ person tent (the best spots are located near the hut). The hut is usually much warmer than a bivy but you will have no view and no privacy in the morning.
- There is still lots of good snow to melt on upper sections (not on the summit though), if you know what I mean.
- Bring an extra bar of chocolate and give it in the morning to hungry JMT-ers (dirtier and skinnier people coming from the West) - you will instantly become the most popular person on the mountain
Edit to add: there is nothing hardcore about getting to the top with overnight gear in one day, but it will be a long day if you come from sea level. On the up side, the full moon illuminates the main train so well (well ... most of it and on clear days) that you can easily finish the ascent under the stars - also a cool experience.