Mutating a SHA-2 input would allow us to have multiple hashes for a single input, would this increase SHA-2's security? Let me elaborate: Say we have two distinct inputs: $A$ and $B$. We also have some arbitrary deterministic mutation protocol $M$ (for example reversing the characters and performing a Caesar cipher).

Say by chance (I understand this is extremely unlikely) $A$ and $B$ result in the same hash when run through SHA-256. However, when $A$ and $B$ are mutated with $M$ first, and then run through SHA-256, they have different hashes.

Am I correct in this thinking? Deterministic mutations would allow us to have multiple hashes for a single input which would make the probability for collisions even lower, because another input would have to have the same hashes for all mutations.
Arbitrary mutation protocol is equivalent to a random permutation (i.e., a random invertible function); so running through $M$ is basically equivalent to encrypting with some random key $K$. You don't need something so complex as a permutation - you could be adding on a fixed prefix, suffix, etc, just as well.

From a hashing perspective, there's no real reason to do this.
Your chances are already $2^{-256}$ anyway for a collision between two inputs (more inputs increase the chance via birthday).  For each change you do and output you're basically increasing the bit-length, so $2^{-512}$, $2^{-768}$, etc. (so you are increasing security). But you could just as well just use SHA-512 if 256 bits isn't enough for you (and it generally runs faster on modern processors)