My understanding is that the card will only supply enough power to a set number of connectors, thats 2 of the 5 on my 7970. Mind I like the way your understanding works better but after looking around there doesn't seem to be much in the way of understanding how the active adapters work since most of them don't have an external power source or much of anything else that says they're active except the fact that they usually convert X to Y and are bigger than the typical adapter of that type that comes with the card... usually.
I'd say since its really only around 30 bucks, and thats small for the price of a 3rd monitor, to try and find an active hdmi to dvi or something like that and give that a shot.
Also after looking at the wikipedia pages for both hdmi and displayport they both put out similar amounts of power. 3.3v @ .5amps for displayport and 5v @ .5amps for hdmi. While sure it is a difference, power output is still part of the connector and will be required of each. Also if I'm not mistaken the power output is more for the signal range of the cable as display port can have an output of up to 16v and hdmi is set at 5v no matter what. I've read theory that the active adapter is more for signal diferentiaton on the card but as I said before, how they work is still up in the air.