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.

You mention that the persons in question have a master's degree.

In Mexico and I guess other Hispano-American countries, someone with a master's degree will be called Maestro/Maestra. Likewise, the formal way to address a person holding a bachelor's degree will be either Licenciado/Licenciada, or Ingeniero/Ingeniera if he/she is an engineer. Anyone with a teaching load at a university is considered a profesor and could be addressed that way. But the title from the degree is usually more formal.