It's sort of a game with amateur tribologists to use something other than the factory recommendation. The Sturmey factory consistently specifies NLGI #2 grease for the bearings and labyrinth seals and either NLGI #00 grease or 20~30 wt. motor oil for the internal mechanism.
Fun fact: during and after the war, when things were rationed and in short supply, Sturmey suggested sewing machine oil as a substitute for the internal mechanism and Vaseline for the bearings and seals. (!)
How about Finish Line Wet Bicycle Lube to be dripped into a Sturmey hub? I can't say how it compares to 20-30 wt motor oil.