Can one group the SHA-256 outputs depending on partial inputs?
Somewhat simplified, if that were possible, you would have found a vulnerability in SHA-256. The kind of attack you're talking about is called a preimage attack, ie figuring out information on the original message from the hash. There have been some attacks of that kind against a weakened version of SHA-256 but no successful ones against the standard algorithm.

Of course your MD5 only adds 128 bits of entropy and your other data one (either all a or all b), but a brute force attack against 129 bits isn't what you're looking for either.