Linen thread, even colored, is pc and stronger than cotton. If you want to move beyond linen then sinew would be an option.
A saddle stitch is where you put a needle on either end of the thread and lace like a pair of shoes. This is pretty much standard with linen thread. The thread can get frayed after a while and to prevent this keep waxing the thread every now and then.
On the sinew, I keep one end dry and use that as a "needle". I make a hole with an awl and then run the stiff end of the sinew through the hole. Since you have real braintan, you may want to consider sinew. With sinew use a whip stitch and when you get toward the end of a piece of sinew, lay the end along the unsewn seam and then feed in a new piece, the end of the new piece gets laid down along the unsewn seam, then start whip stitching again over the ends, locking them in place.
Another neat trick I use is small brads. If you use an awl to make the holes, the holes can close up on you. I therefore make the holes, all spaced nicely apart, and put in the brads. I then pull out brad by brad as I sew. I use this with either linen thread(saddle stitch) or sinew(whip stitch).