You probably have a poor connection somewhere caused by corrosion. With the low current draw of your voltmeter you get a good reading, but under load of the autopilot motor there is a voltage drop. These things can be elusive to find. Start by running a temporary connection direct from your battery bank to your autopilot, to confirm that that's the issue.

It could be a weak crimp, dirty breaker/fuse, dirty switch contact or spade connection. For some reason the issue is usually on the ground side of the path.

Good luck.