A question on the diagonal lemma

The formula $\phi(x)$ to which we are going to apply the diagonalization lemma needs to have, as you say, one free variable.

But if $\chi \land Prov(\ulcorner\chi\urcorner)$ has a free variable at all it has to be in $\chi$ since $Prov(\ulcorner\chi\urcorner)$ is a closed wff, even if $\chi$ isn't ($\ulcorner\chi\urcorner$ is a numeral in every case).

So you really need to write your $\psi$ as something like $\chi(x) \land Prov(\ulcorner\chi(x)\urcorner)$. And you can apply the lemma to that, but we get something quite different from what you wrote, i.e. there is $G$ such that $$\vdash G \equiv (\chi(\ulcorner G\urcorner) \land Prov(\ulcorner\chi(x)\urcorner)$$

Which is quite unproblematic.