Epoxy is the way to go in my opinion, and to apply a light pressure of sort to keep it in place while it’s curing.
Super glue is not a good choice.
If the tenon doesn’t fit, because it’s too big for the mortise, there may be too much gunk in the mortise, or the stem wasn’t made for the pipe.
If it’s funk in the mortise, clear it out, if the tenon is still to big or there isn’t crud in the mortise you can try sanding the tenon until you get the proper fit.
Don’t know if that helps , but that’s a couple of ¢’s in any case.
P.S. There is a chance that mortise has a tapered bore, like the army mount pressure fit.