Extremely strong glues will likely be unsafe for hamsters. Even Elmer's glue/white school glue/PVA type glue is questionable in safety despite popularity(made with synthetic petroleum derivatives). I would, however, consider trying to make wood pieces for example slotted so they just slide together or perhaps a flour based glue(ideally using an unfortified/unenriched flour, brown rice flour can even be an option).
Another idea I've personally had that I would consider extremely safe with some precautions would be a homemade gum arabic(aka acacia gum)and water solution. The precautions here would be to never use the art grade gum arabic as this is not safe and can be loaded with potential impurities and contaminants and isn't food-safe, however a high quality food safe gum arabic would be a great binder and potential glue(for example in seed stick type treats instead of honey, since there'd be no added sugar). I can't take credit for this idea and I do forget exactly where it originated, but from what I do remember I think it had something to do with avian enrichment in Australian zoos(however a good food grade gum arabic is safe for more than just birds).
https://www.baldwins.co.uk/baldwins-...acacia-arabica I did find this source for a food grade gum arabic powder in the UK. The glue itself should only be made with the powder and water(1 part powder to 5 parts water from what I could find), nothing else should be added.