Hi Georgina. In the context of the excerpt, I don't feel it is particularly racist. You tell us Cat is making assumptions based on her unfamiliarity with the not-so-glossy parts of the capital - probably from what she's seen on TV or in newspapers. But, having said that, the word black is a difficult one.
I wonder if, as it's dark in the park and, I assume not daylight bright, could you describe his build first? Along with the dog, that could feel very threatening for a woman alone and already in a stressed and nervous state. Then, when they're introducing themselves and making their apologies, you could describe his colouring (some other way) as a secondary issue. Then you'd take the emphasis off Anthony being black.
If, as you say, he's a friend, his colour shouldn't matter.