[rationalist]

Maybe start with an apology: hey, I'm sorry if I shocked you with that, I know it's not conventional, and the last thing I want to do is make my friends feel uncomfortable. I know that some people can think it's weird because there are some weird people that do weird things in fursuits, but I just wanted to let you know that's not me. It's just a character I like to cosplay as, kind of like how people dress up at anime conventions

Just an example, I don't have all the details or know you or anything. Edit it or write your own, or take a completely different approach. I'm not an expert at these kinds of things.

[rationalist]

If you've tried reaching out already in the past couple of days and they were ghosting you: maybe have a day-long cool down period (not texting them), then say, "I value our friendship, so I asked some (non-fury) people for advice, and I want to apologize...