The limits of 8-inch secondary batteries

Should a very large post-First World War battleship use 8-inch or larger secondary guns?

Usually not; 8-inch guns are too slow against destroyers and offer too little advantage over 6-inch weapons against cruisers.

The first question is what target an 8-inch secondary battery is meant to defeat. The guns lack the rapid fire needed against torpedo craft. Against cruisers, their advantage over 6-inch weapons is too small for the space and weight they consume. A ship large enough to justify them would probably carry main guns in the 18-to-20-inch range. The secondary battery would therefore become another costly part of an already extreme design.