In the event anyone else has problems with a loose mortise insert similar to this Kaywoodie, I repaired the issue. The pipe was an estate find and quite filthy, so after a very good brush and alcohol cleaning, I found that the mortise has threads cut into the wood, and the mortise insert has male threads that match (so envision the insert as a bolt and the mortise as a nut). These threads were so packed with tar that I thought the pieces weren't threaded but smooth.

After cleaning, I simply threaded the insert into the mortise using no glue, as the fit was very tight like a pipe plug. It got to the point where the insert would not fully thread in with finger pressure, so I cut a piece of 3/8" automotive fuel hose in two, wrapped the jaws of channellock pliers with the hose halves, and then tightened the insert all the way in. After all was said and done, I used a retort to do a final cleaning of the pipe and had no leak issues around the insert-to-wood or stem-to-insert joints.