Yes, you can use the TBIII to put the silk down. It depends on the stiffness of the silk as to whether you need to wrap or not. I usually wet the material before gluing(I think it helps to absorb the glue) and if it drapes nicely over the stave, you shouldn't need to wrap. If it won't stay put on its own then a wrap is in order. I use strips of old bed sheets for wrapping. It's cheat, usually readily available and breaths well so the backing and glue moisture can evaporate.
Put a thin layer of TBIII on the bow's back and let it cure. This will be a moisture barrier so more moisture doesn't get into the wood when backing.