If the trees have fully bloomed, you could use a DLI light meter to measure the amount of light reaching the turf under the trees and out in the yard. This would increase as the days get longer, but would give you an idea of the amount of light under the trees vs the amount of light in the open. With that information, you may be able to discern whether a 'shade-tolerant' bermuda cultivar would thrive under the tree in a manner that's suitable to you, or whether a different type of turf grass would be required.