LED controlling protocol like WS2812 for more colors

You could just use the chip inside those Neopixels and connect like two 
or three of them per lamp, giving you 6 or 9 channels.

The protocol of any RGB lamp just transfers (usually 8-bit) brightness 
values, not a color, so youc ould connect any lamp you like.

Hey,
yes, but are there LED strips that have that functionality?

I would want to just have a reel like a normal LED strip, but have more 
control channels for more colors.

I'd say no.
There are RGBW LEDs with one white, and there are Strips with those.
Then you find Strips with CW and WW LEDs, but those lack RGB.
I've never seen strips with those weird colors you want, even normal 
LEDs are hard to get, so you may not even find lamps with that mixture.


Commercially available RGB/CW/WW lamps have separate LEDs for RGB, C and 
W, and i guess there's a reason for this... ;)

There is the ws2811 led driver chip. You can attach any led you like and 
cascade as many as you need. Otherwise I see no other solution beside a 
discrete build. Take a small microcontroller and an led driver like the 
tlc59116 and create your own protocol.