Just did the exact route you're planning. You probably don't need to bring an ax or crampons at all. Here are the details:
The worst snow is above Upper Boyscout Lake and is somewhat firm in the morning. We were able to traverse it no problem in approach shoes, but a slip and fall would be bad. There are good footsteps most of the way and I was easily able to kick steps when needed. The snow next to Iceberg is very soft, sun-cupped, and wet. Once you get past Whitney-Russell col, there's no snow to worry about, including the descent via the East Ridge. There is snow in the top of the gully heading down from Russell towards Upper Boyscout Lake, but it is easily avoided by veering right. Have a good time!