I finally settled on the ASUS HD7970-DC2T-3GD5.
The Corsair HX520W seems to handle it just fine and it had precisely two spare "6+2" connectors free.
The challenge was the Thermaltake Lanbox Lite.
The big(!) asus did not fit. It was:
1) Too wide to pass between the vertical and the horizontal drive cages.
2) Had the two 8pin power-connectors placed on top of the really tall card - clashing against the bottom of even the highest part of the drive cage assembly. The former card had its connectors placed to the rear.
Here you can see just how tight a fit it was, even after removing pieces of the drive-cage!
(Click on the pictures to enlarge, zoom and focus).
pictures.lytro.com/DukeP/stories/57862
pictures.lytro.com/DukeP/stories/57863
(Some of the tools needed: pictures.lytro.com/DukeP/stories/57864)
Thanks all for the advice!
~~DukeP~~