I have a 12V LED strip. It is basically one WS2811 controller chip to three RGB LEDs in series, which is how they drop the voltage for the RGB chips. The 24 bits shifted into the WS2811 is displayed in all three series LED and constitute one address on the strip. Therefore a 300LED strip looks like a 100 LED strip to the Lumazoid code.
I wired this onto my own PhilduinoZoid (and arduino board running the lumazoid code from github) The onboard VR supplied the 328P ( I may have had to remove the power reversal protection diode to get and extra 0.5VDC on the VCC, maybe even the 5v regulator to because it is in 328P VCC circuit when powering from the barrel connector) with 5V. Then I connected the strip power to the barrel connector tab going directly to the 12VDC power supply. The digital drive from the Arduino 328P was plenty for the strip data line and the 12VDC no problem for the Arduino.
Since the factory Lumazoid does not have the Voltage Regulator you may be able to get the with the factory Lumazoid to power a 12V strip.
NOTE: I have not tired this. Perhaps if you had a 5v supply going to the factory Lumazoid (like the USB input) and put 12V to the 12V strip and strapped the grounds you could get it to work.
That’s cool. I didn’t know there were strips with 3 RGB LEDs in series for each address. Good job on all your work.
Regarding your last note, I think you are right: you could provide the Lumazoid board with 5V and the LEDs with 12V, connecting all the grounds of course.