I haven't experienced this exact phenomenon but I am inclined to agree with the Furuno tech.  If I understand your description of the problem the AP gets the coordinates of the waypoint it's going to, turns towards it (and I'm assuming turns to the correct side, port or starboard, each time) and then overshoots the mark, but eventually corrects back to the correct heading.  

I'm making some assumptions but I believe it's highly unlikely the TZT 14 is sending incorrect coordinates for the waypoint and then eventually correcting them.  In fact, I suspect it puts a set of coordinates onto the wire that the AP course computer picks up.  Though extreme, it almost sounds like your describing a case where rudder gain and sensitivity are incorrectly set causing the AP to overshoot the mark, but in this case that's a pretty big overshoot.  

Are the AP and MFD connected via NMEA-0183 or NMEA-2000?  I think you can probably rule in or out the MFD by getting a view of the data exchanged between the MFD and the AP.  If the MFD, as I suspect, is simply transmitting data on PGN 129285 ( https://www.nmea.org/content/nmea_standards/messages_pgns.asp) indicating the next waypoint in the route, then from there it's up to the AP to figure out how to get there and it sounds to me like that's where the breakdown is occurring.  

Ben

When in Nav, the boat always turns in the correct direction, but it continues past the bearing quite substantially.  I have re set the rudder gain to a lower number, no help.  The PGN output is what the Furuno tech suggests; just 2.  Cross trak error, and Nav data.

What is strange is that a turn to starboard is much worse than a turn to port.

Go figure.