Hi Eian, unfamiliar with Axiom, but do you know whether it has an NMEA output data wire for sending GPS routing information? I had a similar problem with connecting my then new Garmin 178C Sounder many years ago. Long story short, the Garmin had a single blue output data wire that sends NMEA routing data to a device, in this case the Autohelm 6000 head. Took several calls to both vendors before I got a fix. The Autohelm has an older NMEA version of software compared to the Garmin, but the 6000 head does receive the data and keeps the autopilot on a steady course.
The only inconvenience, if you want to call it that, is when you come to the end of a waypoint and it is time to change course, depending upon the amount of course change due to the different NMEA versions, it takes a long time for the autopilot to accept and plot the new course adjustment. So, what I do is manually make the course change to the new waypoint. Otherwise, the autopilot snakes a long course to the new waypoint. Once on the new course, the Garmin keeps the autopilot dead on, even making drift adjustments.
No electrical connection to the 7000 course computer other than through the 6000 head, but it does work. Whether your Axiom system has the same, I would suspect so. Wrote an article about my installation and the end result. It took only one wire for one way communications from the Garmin to the head. The head has no output port to the Garmin. That article is here:
Hunter 42 Owner Modifications and Upgrades