When sewing fake fur, I was taught to carefully trim the fur off within the seam allowance & hold/tape the rest back from where you are sewing. This lessens the bulk & ensures that you are getting a solidly sewn seam without all the fighting. Also, be sure to comb/brush the fur on the right side of your project & the seams will disappear.
Good luck. Hope you find a solution you feel comfortable with.
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