I know it's too late to say this but one of the biggest reasons I don't use tite bond 3 over hide glue is because then you can't reuse the sinew if somehow the bow breaks or something goes wrong.
With hide glue you can just submerge the project in water to let the sinew rehydrate and it comes right off. You can then separate the sinew fibers. From there you can either freeze it wet, dry it out, or apply it to the next project.