I have read from other people's posting that they have used various wood glues with the sinew, but I am not convinced that they are going to get all the benefits of the sinew without the use of the hide glue. There was another post recently on using artificial sinew and one of the replies went into detail on how the hide glue and the sinew worked hand-in-hand. After all, they are virtually made up of the same proteins and such.
When I do a sinew job, I finish it out with two coats of hide glue over the sinew once it has dried a few days. It fills what little I get for gaps and makes a nice smooth base for whatever goes on next (rawhide, snakeskin, fresh air). How about doing your sinew job with hide glue and then at the end, cover it with a few coats of TBIII for waterproofing?