As mentioned by Ted, the appropriate generalization is for odd levels only. In Hjorth "Variations of the Martin-Solovay tree" and "Some applications of coarse inner model theory", the following folklore result is stated:

If $E$ is a thin $\Pi^1_3$ equivalence relation on $\mathbb{R}$, then for every $x$, there is a $\Delta^1_3(x,<u_\omega)$ set $A$ such that $x \in A \subseteq [x]_E$.

As defined in these papers, $A\subseteq \mathbb{R}$ is $\Sigma^1_3(x,\alpha)$ for $\alpha<u_\omega$ means there is $B\subseteq \mathbb{R}^2$ such that $x\in A$ iff for $(x,y)\in B$

As mentioned by Ted, the appropriate generalization is for odd levels only. In Hjorth "Variations of the Martin-Solovay tree" and "Some applications of coarse inner model theory", the following folklore result that generalizes Harrington's proof of Silver's dichotomy is stated:

Assume boldface $\bf\Delta^1_2$ determinacy. If $E$ is a thin $\Pi^1_3$ equivalence relation on $\mathbb{R}$, then for every $x$, there is a $\Delta^1_3(<u_\omega)$ set $A$ such that $x \in A \subseteq [x]_E$.

As defined in these papers, for a fixed $\alpha<u_\omega$, $A\subseteq \mathbb{R}$ is $\Sigma^1_3(\alpha)$ means there is $B\subseteq \mathbb{R}^2$ such that $x\in A$ iff $(x,y)\in B$ for some sharp code $y$ for $\alpha$. $\Sigma^1_3(<u_\omega)$ means $\Sigma^1_3(\alpha)$ for some $\alpha<u_\omega$. $\Delta$-pointclasses are defined in the obvious way.

The correct generalization of the Gandy-Harrington topology appears to be generated by $\Sigma^1_3(<u_\omega)$ sets. Also as a corollary, every thin $\Pi^1_3$ equivalence relation is $\Delta^1_3$ reducible to a $\Pi^1_3$ equivalence relation on a $\Pi^1_3$ subset of $u_\omega$ (in the sharp codes); every thin $\Delta^1_3$ equivalence is $\Delta^1_3$ reducible to equality on $u_\omega$.

I haven't read the proof of this generalization. So it would be nice if someone can expand a bit more. I also don't know the proof of the basis theorem at lower level with this approach, even though it seems to be straightforward.