Turn distance, not sensitivity
Every engine has a yaw constant: how many degrees the view turns per mouse count at sensitivity 1. Counter-Strike is 0.022, Valorant 0.07, Overwatch and Call of Duty 0.0066. Those constants are the reason the same number feels completely different across two games.
So the conversion never tries to scale one sensitivity into another directly. It works out how many mouse counts your current setting needs for a full 360, turns that into a physical distance using your DPI, then solves the target game's sensitivity for the same distance.