All I can find is the CC-4000 is a chase controller. Typically, they don't draw much on there own, same as dimmers and switches. I know this has to be confusing since the CC-4000 has fuses and current draw ratings, but this is mainly for current passing through. Since you're on a power inverter with limited capacity, I can certainly see why knowing this information would be important.
What's also important as what are the LED fixtures you're running, as perhaps you shouldn't be running those off a chase controller in the first place.
I did a search for CC-4000 and American D and I found something in regards to the Sparkle manual, which specifically mentions the CC-4000 as a chase controller.
Sparkle Manual