I'd say point her to the multitude of fundraisers for animal charities and others generated by furry conventions.
From what I've seen, furries, in general, tend to have a great deal of respect for most living creatures - Especially the species they identify with.
I'm pretty sure respect is in the opposite direction of the mockery that seems to be implied by your girlfriend's arguments.
If she's equating it to black-facing, I'd just correct her, and point out furries do this because they usually quite like their chosen animal.
I mean it sounds like, by her reasoning, dressing in a suit and tie would violate the rights of all other people who wear a suit and tie for 'copying their image without consent'.
And since when did we need to have consent from the entire population of earth to dress like people? We're all just strange animals playing dress-up at the end of the day anyway.
If all else fails, see if you can convince her that clowns should be considered racist against white people. That might be good for a laugh.
Granted, clowns might be offensive to most people in general, but I think that's for different reasons...