Can an illuminated switch be made to work with LED bulbs?
A dimmer-compatible LED bulb might work. A higher wattage LED bulb could also work better, but most likely you will need a series pass element like the referenced Lutron LUT-MLC in parallel with the bulb. This is 'like' adding a resistor, but it should not get as hot. If you use a suitably rated component, it should be designed to fail safe (i.e. open circuit).

To add a bit more context... The neon indicator passes a pretty small current when it is operating, and has a fairly high but dynamic series resistance. It is quite possible for an LED power supply to extract enough voltage to just light the LEDs (maybe 30v, depending on the design). A shunt bypass element needs to pass most of the indicator current in order to ensure that the LED can't light (and the internal LED discharge resistor should help). Since all LEDs are different, there is no 'simple' answer.