I ordered some from DesignsBySiCK. NOT the Brother set. That is for the Disney characters. But their other set has some good browns.
Colors:
1940 or 1977 for regular chocolate
2235 for a dark chocolate
2147 for a cherry type chocolate
and from the Brother type set
339 for a regular chocolate color.
OK, I guess their site for thread is Embroidery Passion. I'll attach a link. I've also found another site for thread, but have not tried anything except the variegated stuff. But it is really nice too. It is Metro Thread. I'll attach that link as well. Good luck.
Oops, my brain didn't process the "fur" part. If you can afford it, I would recommend getting a whole set of the Brother threads and a whole set of the other threads from SiCK. Their shipping is a little bit tough to swallow, but it doesn't increase very much with additional items in the same order. I bought three boxes of the Brother thread and four of the other thread. If you watch for a sale, you can get it for as low as 60 cents per spool. Then you can make anything with your own color choices. I haven't watched Metro for their thread sales. I got a full set of the variegated threads for 60 cents per spool. The two seem to be comparable in value. The only difference I've found is that Metro has the snap cap spool to secure the thread when it is being stored. But SiCK has their label affixed in such a way that you can always read the thread number. If I would have done it again, I would have had them throw in a "sample" bag of the white and black, since that is what I run out of first, and now I am set for the near future.