Not an answer, but this may be helpful:

**Theorem 1.** If $X$ is a Lindelöf Tychonoff almost strongly zero-dimensional space, then the following are equivalent:

(i) $X$ is strongly zero-dimensional;

(ii) $X$ is *almost zero-dimensional*, that is, $X$ has a neighborhood basis of C-sets.

Proof. (i)$\Rightarrow$(ii) is trivial, and the converse follows from Theorem 4.3 in this paper (we assume separable metrizable there, but Lindelöf should be enough). $\square$

In light of Taras Banakh's comment above, for separable metrizable spaces I believe the question is: *If $X$ is separable metrizable and $f:X\to Y$ is a continuous bijection onto a zero-dimensional space $Y$ which maps open sets to $G_\delta$-sets, then is $X$ almost zero-dimensional?*

**Theorem 2.** Every almost strongly zero-dimensional homogeneous Polish space $X$ is (strongly) zero-dimensional.

Proof. If $U$ is any open subset of $X$, then $U$ is a $\sigma$C-set, so by the Baire property there is a C-set $F\subseteq U$ which contains a non-empty open set. Continuing this process we construct C-sets $F_n$ such that $F_{n+1}\subseteq F^\mathrm{o}_n$ and $\text{diam}(F_n)\leq 1/n$ in a complete metric. Then there exists $x\in \bigcap F_n$, and $x$ has a neighborhood basis of C-sets. By homogeneity, $X$ is almost zero-dimensional, so by Theorem 1 $X$ is strongly zero-dimensional. $\square$

More generally it is true that each almost strongly zero-dimensional Polish space is zero-dimensional at a dense $G_\delta$-set of points.