I'm glad you asked, because I actually did think about that in the beginning. The problems are twofold. First, there are no RGBW LEDs as small as the RGB one I'm using, so I couldn't have made it a 16mm switch. And secondly, instead of 36, you now have to control 48 individual LEDs, which would make each of them a little dimmer and would require a beefier microcontroller, which of course wouldn't fit into a 16mm switch either.
There's also little benefit to RGBW outside of room lighting.
BTW, I couldn't imagine nobody had made something like this yet, and I was right. The
Blinkstick Pro together with the
LED adapter is exactly what you're looking for:
Up to 4A for the LEDs, and you can connect the 12V from the PSU yourself. And what's even better, you can get the LED adapter (the thing on the right) separately, so all you'd have to do is hook that up to my button and you'd be golden.