He appeals to those constructs to argue his case to those who ARE bound by them, who DO believe in them. Yes, he would readily admit that if he cannot know anything, then he cannot know that he cannot know anything. But he's not trying to convince himself. He's trying to convince someone who actually believes he CAN know something, and that logic IS binding.
Putting aside the obvious question of why a subjectivist would be trying to convince some one of some thing, how is it possible to rely on the laws of logic to prove that the laws of logic do not exist? It is absurd. Still, the one who claims that he cannot know anything, (even though he cannot know that he cannot know anything) cannot defend his claim to epestimological incompetence without invoking some objective standard that of necessity requires that he can say something rational about knowledge of the unknowable, which is inherently self-vitiating.