If you want to propagate a non-Pauli operator through a stabilizer circuit, a useful conversion is to transform it into a form where all the non-Pauli stuff is hidden away on ancilla qubits.
For example, a CNOT can be rewritten like this:
Which seems like a terrible idea, but because the parts touching the main circuit area are just Pauli product controls you can now easily move it through stabilizer operations by rewriting the Paulis. For example:
Once you've finished moving it to its final destination, you can attempt to convert back to a circuit that doesn't use ancilla qubits. Or just leave it in this form; it's really quite a useful way to understand a CNOT in a more generalized basis-independent way. It negates the amplitude of states in the intersection of the -1 eigenspaces of two Pauli product observables.
This same trick works for any operation you want to move through a circuit, including non-Clifford operations like Toffoli gates and T gates. It's the same underlying idea behind Clifford frame tracking used in "A Game of Surface Codes".