The friend in question wouldn't buy that argument though, because rather than accepting as a premise that they hold inconsistent axioms, they would assert that they don't apply things like axioms to their reasoning about social justice.

Plus, it's not likely to reflect their impression of their own actions. They're probably not trying to logically derive conclusions from a set of conflicting premises so much as they're following their native moral instincts, which may be internally inconsistent, but certainly do not have unlimited elasticity of output. You can get an ordinary person to respond to the same moral dilemma in different ways by framing it differently, but there are some conclusions that they cannot be convinced to draw, and others that they will uphold consistently, so if they're told that their belief system can derive any result, their response is likely to be "What? No it can't."