Is AES key and cipher distance measurable? Could this be translated into a cost function?
Currently, there is the possibility for only one cost function $C(k, k')= (k \neq k)$ or $C(c, c') = (c \neq c')$. This function shows zero cost for a correct guess and full cost for an incorrect guess


Not quite. Hamming distance is an obvious candidate, as in the comments. Another candidate might be the compatibility of the two permutations $E_k$ and $E_{k'}$ or  $$\#\{x:E_k(x)=E_{k'}\}.$$ There are other distances on permutations such as Kendall tau, as well.

It's probably going to be difficult to do this except at a reduced scale, so I remain skeptical. Maybe try it on weaker or reduced round ciphers.