Just change the switch to a plain switch. They cost $2.
The faulty idea of powering switches
The issue here is an obsolete way of powering "switches which require power themselves to do their thing". That includes lighted switches, motion sensors, day-night sensors, dimmers, smart switches, you name it. In your case the switch contains a snall neon light.
The scheme depends on the electrical behavior of an incandescent (or halogen) bulb: it behaves like a low-value resistor when it is off. The switch only needs a small amount of current, so it leaks it through the bulb. The current isn't enough to heat up the incandescent to its luminous temperature.
But this won't work on an electronic ballast or driver. It will just make the ballast go insane. Even an old magnetic ballast for fluorescents, it's likely to use that current to light the tube pre-heaters, making the ends glow orange. If the fluorescent has a starter, the starter will constantly cycle.
So the "leak current" strategy is obsolete and all those should be boxed up and shipped back to the 1970s.
The modern way is to bring neutral to the switch, so it can use hot+neutral to power itself, like every other electric appliance in the world. Now the switch has a solid power supply, and can work like a plain switch - either connecting the switched-hot to hot, or not. And that allows any load to work.
Your 3-way switch wiring is also faulty
Typically, "unusual" switches are one-trick ponies. You get a switch that's 3-way, OR a lighted switch, OR a dimmer, OR a motion sensor, but a "lighted 3-way" would be two tricks, and they don't make those. So the last guy sacrificed 3-way to get lighted.
It sounds like this circuit was intended to be a 3/4 way switch complex (2/3 way for the British) and the last guy really really wanted a lighted switch, and was willing to compromise away 3-way functionality.
So, since you don't care about a lighted switch, and want to roll it back to a plain switch anyway, you could take the opportunity to restore its 3-way switch functionality, so your one trick is proper 3-way instead of lighted.
Are you sitting down? 3-way switches will set you back three dollars.