Mullet I ve done a very simillar experiment. Just not on a bow.
Try out abot 5-6 kind of D3 PVA glues I could by hear. Put a fat layer on a silicone paper and let tham dry for weeks at a warm, place.
All of tham was pulled with the same weight and with hand too. Some of tham was better, but all of tham stretch like rubber just not turn back for the same size as rubber do.
PVA glues working fine if use it a very thin layer, like between two compressed wood. In a sinewing process U couldn't press all the glue out so it will contain a relative big procent of glue.
One of my learned friend wrote her thesis from PVA glues in paint restauration and she done a lot of experiment. All of tham was dryed out and streched in controlled conditions. She had the same conclusion like me.
The best I find here is a Pattex D3 PVA wood glue for all wood. It was the best (harder and less strechy) in my experiments and in this thesis too. I use this in my work and for bows nocks etc. It is still not enough strong for lamination so I think TB III schould be stronger.
I couldn't by TB glues here
Another important thing! All PVA glues could form with heat. So if U push a piece with hot iron (between silicon paper) it will melt a bit at about 60-80C.
I dont think it is a good thing if use the bow on a warm sunny day