Welcome, Alliefite. I'll post a slightly different approach: Get the proper equipment, read up on everything you can, and go -- with the idea that you can go as far as you feel comfortable, and turn around if you feel you are getting into something that is over your experience and comfort level.
There are many people who successfully climb Whitney in May and early June who go up (and down) the snow slope west of the 99 switchbacks and for many, it is their first and only experience using an ice axe and crampons. For the great majority, they do it all successfully without problems or injury. Other than the usual random hiking injuries, the biggest problems occur with conditions icing up due to cold (early mornings and late afternoons), with people without the crampons and ice axes. The Main Whitney Trail area at this time with the snow coverage is not the extreme conditions where only the most well-trained and experienced would go. It is definitely takes care and concentration, but you won't be perched on ice at the edge of a cliff.
When you go, there will be others on the trail, others coming down, who can tell you about conditions higher up. Learn from them what works (or doesn't). You can make decisions to proceed or not based on how you feel, and what you learn and see. Note that by the time you get to the steeper slope by the 99 switchbacks, you will already have walked a significant distance with those crampons, and you will know by then whether you are comfortable using them. You can even experiment on steeper slopes lower down to see what climbing (AND descending) is like. You might even practice a few self-arrest situations, since you are planning to go slow enough that you will have time to spare.
One comment on your Consultation Lake plan: This early, consider staying in Trail Camp, so you can talk to other hikers about their experiences. You will be pretty well isolated at CL, and maybe this early in the season, that wouldn't be so necessary or ideal.
Be sure to take a marmot-proof food canister. They will be searching everyone for goodies!
