Fleet Air Arm torpedo and dive-bombing roles

Did Fleet Air Arm torpedo bombers and dive bombers compete over which attack method to use?

No. The Fleet Air Arm used the method that worked, and the Swordfish could perform both torpedo and limited dive-bombing attacks.

The Fleet Air Arm used whichever attack method suited the situation and was willing to combine them. The Swordfish is the main example: nominally a torpedo bomber, it could also make a reasonably accurate dive-bombing attack in a permissive environment. Its operational flexibility matters more than a rigid role label.