lo.logic – Is there a Hausdorff space whose “covering problem” has intermediate complexity?

For a “reasonable” pointclass ${bf Gamma}$, say that a second-countable space $(X,tau)$ is ${bf Gamma}$-describable iff for some (equivalently, every) enumerated subbase $B=(B_i)_{iinomega}$ we have $${langle f,granglein (2^omega)^2: bigcup_{f(i)=1}B_i=bigcup_{g(j)=1}B_j}in{bfGamma}.$$

For example, Cantor space is properly ${bf Pi^0_2}$-describable and Baire space is properly ${bf Pi^1_1}$-describable. More interestingly, at the end of this earlier question of mine I gave an easy construction of spaces which are properly ${bf Pi^0_alpha}$-describable for arbitrarily large countable $alpha$. However, these spaces were pretty terrible and I’m interested in finding “tamer” examples:

Is there a second-countable Hausdorff space which is properly ${bf Pi^0_alpha}$-describable for some countable $alpha>2$?

The examples mentioned above were not even $T_1$.