Since half your group prefers no snow.... The one thing that comes to mind is the trail to Merced Lake off the JMT beyond Little Yosemite Valley. I have not been beyond the junction to Half Dome and Clouds Rest, but it is in the canyon, so lower and should be pretty much snow free.
Of course the big problem is getting the permit. You would have to get in line early the morning before your hike to try for a walk-in spot, or maybe get a Glacier Point entry.
The snow in July should not be bad anywhere -- temps are warm, the snow is melting, so there shouldn't be that much for people to walk over. And that is all it is -- something to walk over. It is so dense from the melting that there shouldn't be much sinking in at all.
