Your question can be understood as how to do Homological Algebra over the Field with one Element.
Deitmar, in http://arxiv.org/abs/math/0608179 , section 6, gives an example of what can go wrong if you try to do it sheaf cohomology directly via resolutions...
You might also want to look at his http://arxiv.org/abs/math/0605429 ; in order to construct K-theory of monoids he sets up an analogue of the Q-construction. The Hom-sets in the resulting category are sort of Exts, maybe something to start with...
Durov, in http://arxiv.org/abs/0704.2030 , follows the simplicial approach for commutative monads, of which commutative monoids are a special case
