Because the hinges on fine Doubles are made from not too hard steel, I would recommend the use of a Molybdenum based grease.
Kamilaroy has come closest to it with motor bike chain lube or spray grease, as most of these have Moly in them.

For metal to metal sliding action, this is the best stuff, because it prevents galling even if the grease component has completely dried up. It's is used a lot in Air Rifles as well.

You could probably make your own by mixing Superfine Moly with Lithium grease.