Not familiar with that particular VGA cooler, but if they claim compatibility with most GPUs, it should be okay. Sorry, yes, the 6770 takes a little more power, but not a whole lot.
As I mentioned I have attached case fans directly to GPUs before. Most cards recognize that when there is no fan rpm that an alternative cooling method is being used.
At least with Nvidia cards the screw spacing hasn't changed much over the years. AMD tends to switch between a more standard configuration and turning their GPUs at 45 degree angles. But taking a GPU cooler from another card is also an option.
I have seen people drill holes through shrouds on larger cards and attach fans as intakes. I've taken all the fans off of a dual GPU card and replaced it them with a pair of 120mm fans. Used some screws and washers to bolt into the heatsink where the old fans were and zip-tied them to form a rather cool looking arrangement.
Those smaller GPU coolers like the one in the video are a bit trickier, since they don't offer a lot of ways to attach a fan. Industrial doublestick tape with a high temperature rating might work for that. Or a few dabs of epoxy at each corner.