Have you checked out hide glues yet? Hide glue is a type of adhesive made from boiling animal hides. It has a pretty long history. Titebond sells some under their brand, and Old Brown Glue is also a pretty respected brand. It’s a bit more expensive than Titebond 3, but that’s not too bad imo, especially since it’s natural and relatively easy to find. You could try using granules instead of the ready-made stuff, but it might not be worth the hassle for a one-time project. Plenty of information on usage on yt if you want to look into it, the search term has a exceptionally high percentage of old geezers which is interesting but also expected.