The mountains of Arizona (as well as Carson/Santa Fe) will typically hold snow in the higher elevations into May. But it also gets warm in lower elevations by then. May in Sedona averages 85, June almost 95. But that's at 4,500 feet elevation.
May is usually beautiful around 7,000 feet (Flagstaff, Mogollon Rim). Baldy will likely be at or near freezing overnight with lingering snow drifts above 10,000 feet. Abineau - Bear Jaw is rarely backpacked because it's a relatively easy day hike. I can barely think of a place to set up camp that isn't on the Waterline road or the trailhead! There is no camping permitted in the Inner Basin anywhere or above 11,400 on all of the SF Peaks, so it limits the backpacking options in that area. The Cabin Loop is nice at that time of year, but doesn't really have that alpine feel you seem to be going for.
You could use the Kachina Trail to Weatherford (both on the south slopes so snow melts quicker in spring) and there are a few spots to camp before getting too high up. Weatherford once it reaches the Inner Basin can have snow into June and July. It varies greatly depending on snowfall though.
Here's an early May Kachina Weatherford loop I did a couple of years ago: [ photoset ]