I don't know about dampening screws on pickups as I don't think I ever saw them; they use springs on each screw to keep them pushed up and reduce screws play, but I'm not sure they're the same thing.

Anyway, guitars with a spring vibrato bridge, have this faint but audible natural spring reverb inside their body which by contact affects the strings too adding more warmth to their sound.

It would be worth experimenting with a set of all different springs so that the force applied to the bridge is the same, but they would resonate on different frequencies therefore making the reverb effect bandwidth wider.