Dei valet button

Hello there and looking for some help on a valet button. So I wanted to use a dei valet button to turn on a led in my glovebox. Weird I know but I have no lighting in there and was thinking a simple clean way to do this. When I open the glovebox the light turns on simple. Well, didn't think but the buttons are NO and I need a NC button. My ? Is, is there a way to still use the button (because it mounts and fits perfectly) by maybe the way I wire this or use a relay to switch? Or do I need to locate a NC valet button if they make such a thing that same size? Thanks for any info on this.

I can think of one solution, but it isn't perfect.

Use a 3volt LED. You will need a resistor in series with the LED to drop the 12V system voltage to the 3V for the LED.

(-) of LED and one lead of button to GND.
(+) of LED and other lead of button to one end of resistor.
Other end of resistor to (+) 12V supply.

When the button contacts are open (glove box = open) the LED is in series with power through the resistor and ground, and = on.
When the button contacts are closed (glove box = closed), the LED is shunted by the button, and goes off. The button is a now a direct short to ground, but the current is limited by the resistor. There will be a draw in the off position, but with careful component selection, the draw will be negligible in a car that is driven at least once a month or more.

Try to pick an LED / resistor combination with the highest required resistance. I saw a thread on here where one example was a 5mm led that used a 780ohm resistor. The current draw in the "off" position with 780 ohms would be .015ma. (.015ma would take 138 days, over 4 months, to drain 50 amp hours from the battery. Most cars will still start after this amount of drain). The higher the resistance, the lower the current draw in "off".

Not a super-great solution, but workable. Especially if the button fits.