I use an ordinary HP automotive grease - just a light smear (I use the same in winches as well).
It does sometimes result in the clutch needing to be quite tight to stop slipping under high loads but that only for the first couple of trips up and down after greasing. But is not a problem because for its own good I never let the windlass
hold against anything other than light loads.
In my experience it takes a long period (order of weeks) of having the clutch done up very tight for the cones to jamb into the chainwheel. If it does happen I always find that loosening the clutch off and operating the windlass
with the anchor
hard up in the bow roller breaks the cones free of the chainwheel. But that is an infrequent occurance as when not in use I just lightly tighten the clutch and restrain the anchor
on the roller with a strop back to the toe rail (which should be done for safety
even if the clutch is done up tight).