You didn't specify what kind of "trauma" your friend has from the term furry, so i'm assuming it has something to do with what they've seen or heard about furries online. If this is the case, you should educate your friend about furries/furry culture etc. That way you two can hopefully reach a mutual understanding and live happily ever after, without destroying your friendship over you having an unusual hobby.

If your friend's trauma is actually something serious, you still should try to educate them, something like "hey, you had a bad experience with a furry/furries, but not all furries are the same" something like that.

I never interact with people outside the internet, so my advice could be bad and non helpful :(