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A/A Fuel GTX:
Following are answers to some of your questions. But first, if I were the customer I would not tolerate a new installation of a device that flashed or flickered when it is supposed to produce a steady light. Also I am not sure why you need a lighted switch for this application.
From post #1, "The lighted switch seems to be pulsing when the LED is off. I'm assuming it's because of the lighted switch going through the LED circuitry for the neutral connection."
I assume your lighted switch is a two terminal device, and that when the switch is in the off position the switch is to light so it can be found at night in the dark.
In this case there has to be a path for current flow to provide power to this indicator light. That path has to be thru the load. It also may require some minimum voltage to turn on. A sample I tested required about a 90 V RMS sine wave. I might have expected a somewhat lower value because an NE51 initially ionizes at about 70 to 90 volts and glows at about 50 V. Less than the about 50 V level and conduction stops. Note: a 90 V RMS sine wave has a peak voltage of about 127 V. Neon bulbs or other ionized gas tubes must be supplied from an approximate constant current source, or after ionization occurs a very large current will flow and burn out the tube. About 1 mA produces good output from an NE51. We can expect your light switch to require something in the 1 mA range.
Still in post #1. "Is there any reason to be concerned, like will this damage the lighted switch to the point where it no longer illuminates"
This won't hurt the lighted switch whether it is a neon or LED indicator. Quite unlikely that it is an incandescent because of the short life, and some other reasons.
From post #3. "I also assume that this scenario will not adversely affect the LED itself?" fixture.
For a lighted switch you want the least power dissipation when the switch light is on, and therefore the current at 120 V is very low. The current from this lighted switch is so low that it won't damage any normal load that would be attached to this switch, your LED illuminated display. And it won't make that display light. That the current pulses is of no consequence.
From post #5. You would not want to put a resistance across the switch terminals. Instead you would want a small added load across the normal load (Your LED display). Small load means not much power, therefore, a moderately high resistance or reactance.
Do bench experiments with your lighted switch and various loads to see what happens. Make some current measurements using a 1000 ohm 1/2 W resistor and measure the voltage across the 1000 ohms. 1 V equals 1 mA. This resistor will reach its power rating at 22 V, or 22 mA. Resistors are inexpensive compared to meter fuses. Thus, using voltage across such a resistor to measure current where you might accidentally apply an over current is a better solution than using the meter current range.
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