I think this depends heavily on the expectations and trust that A has with respect to C. $A$ should be safe to now view $P_B$ as authentic provided that $A$ trusts that:
- $C$ will only sign $B$'s message if it could successfully verify $S_B(M)$
- $C$ would only use $P_B$ if it had previously obtained and verified the authenticity of $P_B$
- $C$'s requirements for verifying the authenticity of $P_B$ are at least as strict as $A$'s
Note that this is essentially the same as with certificate authorities, where $P_x$ are X.509 certificates (e.g. $P_C$ is the root certificate for a CA, $C$). $B$ can't expect to provide $P_B$ to everyone it expects to communicate with so instead it establishes trust with $C$ which then signs $P_B$. When $A$ receives data from $B$, it would first receive $P_B || S_C(P_B)$ then $M || S_B(M)$. $A$ then decides to trust $P_B$ on the grounds that $C$ would not have signed $P_B$ without first establishing it's authenticity. $A$ then, in turn, trusts that $M$, in fact, originated from $B$.
However, it's important to note the $A$ having authenticated $P_C$ is not sufficient. Messages signed with $S_C$ may, in fact, be coming from $C$ but that does not mean that $C$ is fully trustworthy. For $A$ to trust $C$'s trust of $P_B$, $A$ must also trust that $C$ is sufficiently cautious about establishing the authenticity of $P_B$. Just as it would be extremely concerning if a trusted certificate authority were to start signing any certificate sent their way, $A$'s confidence in the authenticity of $M$ and $P_B$ must not be greater than $A$'s confidence in $C$'s process of establishing trust.
Similarly, if $C$ viewed its job simply as marking that it received and relayed the message, blindly signing the entire payload it received, $A$ would be mistakenly assuming an assurance not actually provided by $C$.
Long story short: $A$ can only trust $P_B$ if it trusts that $C$ is, and is sufficiently capable of, guaranteeing the authenticity of $P_B$.