I don't think Dick's civilian identity is close to Clark Kent or Bruce or even Diana Prince at this point, but it's sort of in a weird area where they know who Robin is and what his role is. And most of that comes from Dick's forays into other media, so they know him on some level. I guess if anyone was to name a civilian ID for Robin, his name has the highest odds of being named due the Adam West show and BTAS at the very least being so prolific, not to mention some of the recent stuff.
As for their track record, post-Snyder most of their films have been pretty solid to great with WW84 being the only major disappointment. I don't trust them wholesale (not by a long shot), but I wouldn't automatically dread any upcoming movies either.