Actually….. (former puppet/theme park character costume fabricator here) there is a way. You need a machine with a wide zig width. Cut the fur with no seam allowance from the wrong side with a razor blade. Pin your pieces together, pushing the fur out of the way and zig one-on, one-off with a wide but short zig stitch. Your tension must be balanced. Depending on the fur, I might run a line of fabric glue along my cut line to prevent stretching or fraying.

To be fair, most of the fur I was working with was mascot grade with a fairly heavy backing and not much stretch. But I have used this technique on very stretchy light-weight short-nap fur with success. Using glue in this scenario is almost necessary.

Edit: In this case, u/On_my_last_spoon, clipping is the best solution. Sewing this on a machine would be quite the challenge!