High-voltage frequency divider
A better and simpler way, in my opinion, is to alter a value of a component in the electronic ballast. The ballast circuit (usually) rectifies the incoming AC power to fuel a power oscillator that then drives the fluorescent tube. Here's a simple picture that should help in understanding: -



And below is a block diagram of one (this one has dimming facilities but ignore that: -



So, I would begin to look for EB circuits on the web and build one just to get a prototype then experiment with altering the oscillation frequency. Then replace the EBs in the lamps with the modified EBs

BUT PLEASE BE CAREFUL BECAUSE THESE THINGS CAN BE LETHAL.

The approach of using the existing EB and inserting something betweeen it and the lamp is basically using two EBs in series and the 2nd EB has to be developed in exactly the same way as what my preferred route is (above).

You also might be able to get an existing EB and alter its operating frequency a bit BUT without a circuit diagram you are somewhat in the dark (pun intended). I'll repeat what I said earlier: -

PLEASE BE CAREFUL BECAUSE THESE THINGS CAN BE LETHAL.