I used to overhaul my 1966 Bendix regularly. It's been gone about 14 years (and its successor quit last year), but here's what I remember:
It's not a fluxgate compass, it's a simple magnetic one. There should be a red lens on the top of the autopilot binnacle. If the light is working you can see the glow through this lens in a darkened room. Turn it on at the dock and check to be sure it has a steady glow with no flickering. The connections to the bulb are not that good and may need cleaning. Inside the binacle a faceted prism floats in a liquid. Fluid may be low, but a bubble didn't seem to bother it much. I don't remember what the fluid is, but smell it to see if it is alcohol. Above it is a right-angle lens that focuses half the compass light on a photocell. If the boat goes too far in one direction more light hits the photocell, and the pilot turns the boat to compensate. The other way less light hits the photocell, and the boat turns the other way. Be sure you have the right bulb because brightness matters. I think the sensitivity control merly changes the current to this light. Start with the sensitivity in the middle of its range.
The compass gimbals are soft brass screw-in pins, half round, half flat, and the rocking of the boat wears them out. You can solder stiff copper wire onto the stumps if they are worn down. Sometimes you can find new ones. A stainless non-magnetic screw would do in a pinch.
There is a hydraulic follow-up cylinder connected to a resevoir near the main steering cylinder. It turns the binnacle so the pilot knows what is happening. This leaks. This fluid doesn't connect to your main steering cylinder, so it can't screw up your steering. Put a pan under it and pull it out, after loosening the stopcock under the resevoir. There is a single o-ring inside the follow-up cylinder which is easy to replace. There will be a sort of clamped seal on the outside end of this cylinder which wears out. It's not important, so ignore it (it just keeps dirt out of the follow-up cylinder). Inside the resevoir is a steel handle. This pressurizes the follow-up system and preloads a spring which is attached to the piston. Fill the resevoir with Type A ATF (automatic transmission fluid) with the handle out. Put the handle in and push down, which you have to do to put the resevoir cap back on. Turn the little stopcock beneath the resevoir to close it and retain that much pressure.
I got 27 years out of the Bendix, so I'm not complaining. Only 13 out of the replacement Robertson. The newer ones are easier, better and more reliable, but you can't fix them without sending them to the shop. And Robertson-Simrad only service two generations of their autopilot back.
Good Luck!!