Personally, I don’t even try to clean black PCBs. They’re like a black finish on a car: anything I do shows, no matter how careful I am. Unfortunately, I don’t have a good solder recommendation as I stocked up a few years back on solder and what I’m using (quite successfully) is no longer available. But look for the non-clean varieties.
If you go to, say, Digikey, you can actually search “solder no-clean” and turn up both leaded and unleaded solders. Digikey uses the phrase “no-clean” in their descriptions; e.g., “SOLDER NO-CLEAN 21AWG 63/37 1LB”.
