There will be a wonderful solutions to you. That is Cyanoacrylate adhesive glue
, it is skin friendly. So try this type of glues.
Cyanoacrylates are superglues, but they give off acetone, which I wouldn't want for any bird. Not sure what's left after they dry, though.
The "skin friendly" versions (2-octyl cyanoacrylate) degrades more slowly than the older versions, so it remains below the skin toxicity level in humans. It's not non-toxic, it's just less toxic.
Wood workers still use horsehoof glue and fish glue, You can buy either in wood working catalogs, or make your own from hoof trimmings, beef bones, etc. The hooves of healthy horses need trimming, so they don't have to die to contribute to the glue factory.