I think when you say "evidence in favour" and "evidence against", what you mean is "data points that are predicted by the standard models of climate" vs "data that does not fit the model" or "data that would require us to change our models in such a way that AGW is thrown into doubt". So when you look at record-breaking years, you're saying "is evidence of AGW", but what you really mean is "this data is predicted by the standard models". Meanwhile, when you say that an unusually cold year "is not evidence against AGW", what you mean is "this data is still predicted by the standard models".
I think the cognitive bias is actually cognitive dissonance, if anything. You have this hunch, that the data doesn't invalidate AGW, and you assume that the existence of this hunch implies "good evidence" or "bad evidence". After all, that's the only thing you can think of right now to explain why you have this hunch. It must be bad evidence, or else you would be wrong to have the hunch. In fact, the hunch is more subtle than that: it's "fits model" or "doesn't fit model". Both record breaking years and non-record breaking fit the model for climate change just fine, and that model predicts AGW. So neither set of data should change our minds. Only data that doesn't fit the model should make us question whether the model is accurate. I believe that this is what you're trying to articulate, except because your mind has gone to "evidence" instead of "fits model", you're questioning your entire thought process. Your thought process is flawed if you're talking about evidence, but if you're trying to figure out whether or not the data fits a model, it makes perfect sense.