lighter grease is better due to the moving parts / pawls etc. Sturmey grease is very thin, also advised that the Shimano lime green internal hub grease is fine to use.
In an steel hubbed AW3 that has become sticky/slippy I have filled the hub with thin engine oil ran the bike round the block and then let it drain out from the port for a week or two. Then I added some thick 140w ep gear oil. Hub ran fine.
Wonder if marine wheel bearing grease or lucas xtra heavy duty grease would be better.