Yes you need more fuel getting to the cylinders when the engine is cold. K-jet accomplishes this by reducing the pressure on top of the plunger which is moved by the air flow sensor plate. This allows a greater deflection of the plate for an equivalent airflow, hence a richer mixture. As the regulator increases pressure, the plate deflection is reduced, and the mixture strength reduced to normal hot values.
So have pressure gauges installed and check the pump for both pressure and flow. Accurate diagnosis is not a guessing game - it gets too expensive.