But I wonder whether you could manipulate them this way arbitrarily far from rational behavior

Surely I can construct such a model. But whether this is generally the case depends too much on the details of the implementation to give a complete answers.

whether they may notice this as some inferences are more likely to be detected when you already have some other facts.

... especially logical inferences: logical deductions of true facts are true, even if you don't know/remember them. But then again, that depends too much on the implementation of the agent to have a general answer, in this case also its computational power would matter.