Let's take into consideration an application which applies a constant voltage to an ultrasonic transducer which is driven at the resonance frequency via frequency control around phase.

What is the important crystal specification that would allow you to achieve performance over a threshold in terms of output power?

Firstly, I recommend having some lenient tolerances around what is considered normal for all properties including dielectric permittivity, compliance, electromechanical properties, and loss coefficients. 

Now, assuming that your crystals are "normal" statistically speaking, the mechanical quality factor measured at high power would be the most important single parameter which signifies high power performance.

Sadly, the measurement of this parameter does not have a industry standard method with standard equipment from big name electronics vendors.