Sorry to revive this old thread, but I recently came across a rope that is similar in stiffness to the training halters sold by some companies, including Clinton Anderson. It is starter rope for small engines. I'm sure this stuff is also carried under different names by suppliers, but I haven't yet determined what they are.
Anyway, a #8 (9/32") diamond braid starter rope can be bought online for around $40 for 200'. You might be able to find it at a local hardware store or a small-engine mechanic shop. 200' would make about 8 or 9 halters, depending on the size. It is sufficiently stiff that it knots well, for the training knots on the nose band, and will stay tied. Being a starter rope, it has sufficient strength to handle a horse. The downside is that they either come in white or a red/black/white braid. Once I figure out just what kind of rope "starter rope" is, I hope to be able to find a supplier that has it in various colors and diameters.
Just FYI, I had a horse recently pull back and break a Clinton Anderson training halter, so I know they are not made of super strong material.