How should staff address non-doctorate professors?
as our institution's culture considers first name acceptable for staff above you in the reporting chain


First name it is then. Or you could go first and last name without a title. Or you can drop the name completely and just get on with telling them what they have to do. It sounds like you know they are all professors/faculty, which would mean using professor is not incorrect, although it is more formal than just the first name and the same as calling someone doctor.

I would steer away from Mr/Ms/Mrs for a couple of reasons. First, it introduces the gender issue you raise. Second, Professor is a higher honorific than Ms/Mr. Finally, working in a medical field where half the clinicians have a terminal masters and half have a terminal doctorate, I find it mildly insulting to use different titles for two people who have terminal degrees.