A detailed study of the pump-intensity dependent characteristics of an all-inorganic CsPbBr3-in-Cs4PbBr6 quantum dot (QD) color converter are reported. This is an attractive material to color convert UV/blue GaN optical pump sources for digital lighting applications, e.g. visible light communications (VLC). It demonstrates narrow spectral emission (522 nm peak emission and < 20 nm full-width-at-half-maximum), invariant with the pump power density (from 0 to 7.15 kW/cm2). The optical bandwidth increases, from 10 MHz at 300 µW/cm2, up to 22 MHz at 3 W/cm2, and 41 MHz at 7.15 kW/cm2. This acceleration of the emission is ascribed to both an increase of the radiative decay rate in the low-pump-density regime, and to the emergence of non-radiative pathways at higher pump density. The higher bandwidth at high-pump density enables a 30% increase in the data rate of a free space VLC link using this color converter.