Thanks @coop. I found a forum post that suggested increasing the X-Plane Flight Models Per Frame, and I increased that to 4 and it seems to have largely solved the problem. It did look like there was something quite laggy going on, which is weird as I'm getting 60fps in X-Plane.
I totally agree that it's operator error. Frustrating that I can't understand or figure out what's going on - especially given I have a lot of experience with other aircraft. My original issue was ILS Autoland not working as described, then a mess trying to solve that. Maybe I'll come back to it after giving it a rest. I can also fly it manually and use autopilot for flight phases other than landing. Thanks.