Most replying on here are against using sinew on your bow, and all their answers are perfectly correct, in my humble opinion. You've got the benefit of some really experienced bowyers who I respect.
However
if the runout is not too bad, and as you say you want to try sinew backing, I personally would give it a go, not withstanding that it's time consuming a bit.
Everybody should sinew back a bow once in their life and this may be your time
It can give you a servicable bow that is definitely all your own work and a learning experience.
However you should probably follow the great advice you've got