Can an oracle allowing errors be non-relativizing?
When considering variations of a computation model, let alone one that is not natural and might confuse our intuition, definitions are of the essense. Your oracle might be in #P but that doesn't mean that is representative of that class. Since you have changed the definition of the function the oracle computes, you will have to either prove the new problem #P-complete or find another appropriate class for which the problem is complete. 

Fooling around the complexity zoo, it seems that such a class has already been defined: http://qwiki.stanford.edu/index.php/Complexity_Zoo:M#modp and http://qwiki.stanford.edu/index.php/Complexity_Zoo:M#modkp , depending on how you choose N.