The general rule for circular orbits is that the escape velocity is √2 times the orbital velocity.
At no point on the esacape trajectory can the object's speed fall below √2 times that of a circular orbit at that distance (or else the object would not escape.) At whatever distance you decide to set the controls for the heart of the sun, you must kill its angular velocity with respect to the sun (because, if it has more than a slight angular momentum, it will follow an elliptical orbit that goes around the sun.) Therefore, at every point on the minimal escape trajectory, the delta-v to redirect the payload into the sun is the escape velocity at that distance. With your strategy, the cost of sending the payload into the sun asymptotically decreases towards the cost of sending it on an escape trajectory.
