There was a time when some cryptographers considered it infeasible for an attacker to find a P and K so as to produce a given C through AES, and there were attempts to build hash functions from AES based on this assumption. But, hash functions based on AES did not turn out to be as secure as other hashing algorithms (see Why AES is not used for secure hashing, instead of SHA-x?).
What you are asking is: given P is it possible for an attacker to find a K so as to produce a given C. This is at least as challenging as the above case, because in this case the attacker is given a particular P, whereas in the above case both P and K can be chosen by the attacker.
So, the answer is: it depends on the resources available to the attacker. If the attacker has sufficient resources, it may be possible to find a key K that produces a given C from given P (or P'). But, it's going to require a LOT of resources.