Anything with chemical adhesive should be avoided, since rodents tend to eat… everything. XD But you could try looking online for some natural/homemade adhesives. I think elmer’s school glue would probably be alright too? It’s nontoxic and meant for little kids’ craft projects. You shouldn’t take my word on that one, though, and should look into deeper before trying it. And of course use as little as possible of whatever you’re using, since too much of anything can be bad.
You could potentially sew or tie it together too. It would be more involved, true, but yarn or strips of cloth might be safer and less tempting to eat. At most, the mice might shred it/tear it apart and use it as nesting materials.
Take a third tube, cut a slit up one side, and put it around the spot where the other two tubes meet. That should hold them together pretty firmly.