If an object knows that it exists, then this implies that it actually exists. Moreover, assuming that the state of a brain is a mathematical fact about the mathematical theory, then that the object knows it exists is in principle a mathematical implication of the mathematical theory (if observation 2 is correct). Hence it would be an implication of the theory that that theory describes an existing reality.
I think your mistake here is that:
It can be a property of a mathematical theory that particular objects, if they existed, would know they exist.
It cannot be a property of a mathematical theory that particular objects exist.
This renders your paradox moot.
If an object knows that it exists, then this implies that it actually exists. Moreover, assuming that the state of a brain is a mathematical fact about the mathematical theory, then that the object knows it exists is in principle a mathematical implication of the mathematical theory (if observation 2 is correct). Hence it would be an implication of the theory that that theory describes an existing reality.