What do you consider to be the optimal AP shell weight & muzzle velocity for WWII-era BBs? I am asking about AP shell weight & muzzle velocity, not any other design characteristic (although, obviously, shell weight effects shell length, BC, SD, et cetera).

Note that for naval guns I look at shell weight in terms of a 'proportion of diameter cubed'. For example: 16 cubed = 4096 & 4096 x 0.5 = 2048; thus a 0.5 'proportion of diameter cubed' 16" shell is 2048 lbs.

For reference, here is the 'proportion of diameter cubed' for the BB AP shells (including some planned but never entered service) of WWII:

0.581 France 1962 lb 15"

0.554 Germany 2271 lb 16"

0.523 Germany 1764 lb 15"

0.578 Italy 1951 lb 15"

0.542 Japan 3219 lb 18.11"

0.535 Japan 2249 lb 16.14"

0.541 Japan 1485 lb 14"

0.580 UK 2375 lb 16"

0.500 UK 2048 lb 16"

0.574 UK 1938 lb 15"

0.579 UK 1590 lb 14"

0.659 US 2700 lb 16"

0.547 US 1500 lb 14"

Muzzle velocities range from a low of 2300 fps for the US 16"/45 firing 2700 lb shells to a high of 2789 fps for the Italian 15"/50 firing 1951 lb shells with the majority being in the 2450-2650 fps range.

For a given gun, lighter/faster shells have inherently better belt penetration than heavier/slower shells due to their flatter trajectory & heavier/slower shells have inherently better deck penetration than lighter/faster shells due to their more arching trajectory. BUT you can go too light/fast or too heavy/slow...

Obviously the answer is a compromise that depends on many factors, including the range at which you expect combat to occure at.