Had you said they were sheepskins to begin with I would have told you there is a 98% probability that they were tanned. 'Fur rugs' generally refers to bear skins, cougars, etc.
If you do it the SAME every stitch you will get consistent results. Whether you run the left hand needle through first or the right hand, do it the same each time.
I think I would go with either straight saddle tan, or dark brown. Perhaps cordovan. When I look at it my eyes should be drawn to the inlay, and they are not.
Look into vg tan sheep. Muc thinner and far more supple than beef. You may want to 'break it in' a bit, but I use it for linin holsters and it is very 'bendable/foldable'.
Or kid skin.