I recommend the LaBelle gear grease. It works metal to metal, metal to plastic or plastic to plastic really well. And you don’t need very much at all, maybe a drop or two. Works like a charm for me.
From solely a mechanical standpoint with respect to those metal gears, the moly-lube would be the best in the long term. However, if its base is an oil that is not compatible with plastic, and you can’t seal the mechanism from oozing or spreading grease, it would be safest to use the grease that Tom recommends and for the reason he states. He has lots of engines, fine ones, and has found how to take care of them so that they would last.
Be sure to clean out the old grease residue. Grease is essentially an oil held in a molecular “sponge” of soap, and released by heat. That hardened gunk is mostly the soap base.
(Moly greases and Teflon greases have additional dry lubricant, which is carried in the grease base.)