Q: If an employer asks you to write a que 'in the style of' a different composer (either living or dead), how do you generally feel about that?
A: They trust my skills, and that I can craft something that will be tasteful and associative to their audience.
Q: Is it a disservice to the composer you're being told to emulate?
A: Don't be silly - of course not.
Q: Is it a disservice to you?
A: No
Q: Is it just fine; part of the job?
A: You should expect this more than not; at least until people call you for being you, then they'll ask you to write something like you previously did. Copy + 5-10% variation = latest craze.
Q: Do you need/want permission from the composer you're being asked to mimic?
A: No one ever does this - ever. Can you imagine someone calling up Hans, "Hey, those Batman jams were sweet, Ima gonna rip you off, we cool?"
Q: Or do you stand your ground against your employer, telling them that mimicing what someone else has already done isn't the way to go?
A1: Depends - if you are independently wealthy, then go ahead and tell them you have a better vision for the media project they dreamed up/went into debt finding funding for, produced/directed/acted in/edited/programmed/fought constantly for/worked 80-100 hours a week on for the several years of their lives on.
A2: If you aren't independently wealthy OR you and and want a career in composition for media, then use their brief as a guide to help you become more creative within the restraints of the project.
Q: Is that a 'hill worth dying on'?
A: Sounds like you are looking for justification now. Do what you want.
Tangent alert: If you went to Uni - you'll be best served to forget the Mozart model the professors worshipped and realize the survivors in this (and any) industry are craftsmen, not 'geniuses'.