You'll need the original PC ROM for the card if you want to rebuild & reflash the OS X EFI into it. Dumping the current one and reflashing would only put you back where you started.
But I believe the reason you're getting a black screen is because the legacy support has been ripped out of ML for the 5770 (possibly starting with Lion). Obviously you're using a PC card which complicates it, but the OS X card is definitely capable of running everything up to Mavericks. To do it you may need to inject the kext and update the device ID.
Did some searching and it seems there was a topic here on NW discussing support. Maybe it has details which can help http://www.neowin.ne...-mac-pro/page-3 and there's also a lengthy topic on MacRumours discussing MP2.1 support: http://forums.macrum...d.php?t=1598176