Just having an ice axe is not good enough. you really need to know how to use it - and that means more than a few hours of trial and error.

Ice axes are great tools for experienced mountaineers. Less so for backpackers. Doing a self-arrest with a full pack on your back sometime is not even possible.

It is hard to say what condtions will be like in July 4. Too early to tell. You could just go with the idea that if you encountered snow you would turn around and retreat. Early July is more "post holing" snow than ice or hard surface snow. (except for early morning- the snow softens by noon). Regardless, you really need to have experience on snow before you do this route early season of anything but an extremely low snow-pack year.