Squarewave modulate CFL

The inverter must free-run to produce a stable output, so you can't modulate it at all. What you CAN do is to shunt current around the tube into a secondary channel of some sort. The drive is probably AC, so you'll want to use a diode bridge to make a DC switch point, and the easiest device to switch these low currents and high voltages might be a vacuum tube.

There will be severe difficulties in the under-1 kHz range if you actually quench the arc inside the lamp, because initiation times are uncertain. The inverter frequency will beat against your modulation, too.

Neon lamps light at lower voltages, and can be run on DC, so are often used for this kind of modulated light source. I've seen neon lamp/rotating disk devices that do realtime Fourier analysis, so you could calibrate a musical instrument, about 1950 era.