I'm not sure why you have an aversion to the so called 'generic' coolers. It's not logical to factor out good heatsink options just because they support a lot of different cards. I may be wrong in this assumption but based on what you've written, it does certainly seem so.
A Twin Turbo Pro will fit based on this
silentpcreview.com | View topic - Twin Turbo Pro Installed on my 5870 :-). I also think a Accelero S1 would work in this case too with cut fins too. Supposedly a 5870 does not need any modding with the Twin Turbo whereas the 5850 is slightly different requiring some fins to be cut.
Edit: "As for your problem with high vrm and mem temps, it is easily solved by taking apart the stock cooler and using the baseplate along with your choice of heatsink"
Seems I was wrong about that, the 5870 baseplate is NOT separable. However, I still think the TTPro is your best option since you can use the provided heatsinks for mem/vrm