Consider any ket. Find the perturbative correction to that ket. Then,
|n> = |n0> + |n1>
Here, |n0> is the ket from the unperturbed hamiltonian (who cares what it is), and |n1> is the 1st order correction.
Do you introduce a new normalization when you add the perturbative correction? It seems the sensible thing to do.