Why is a solution concept of Bayesian Persuasion a subgame perfect equilibrium?
Here are my thoughts. Because the sender commits to a disclosure policy, how you stipulate off-path beliefs doesn't matter (phrased loosely). That is, the modification to PBE is trivial: any optimal persuasion policy corresponds to a class of optimal persuasion policies in which any belief is assigned to each off-path message.
Moreover, you could also resolve your off-path message beliefs issue by assuming that the cardinality of the message set is so small that no off-path messages are sent. This would correspond to, e.g., there only being two messages available in the canonical prosecutor-judge example from KG.