Nice refinements exist and are regular but not Lindelof

Construct the required continuous chain $⟨M_{\alpha }:\alpha <{\omega }_1⟩$ explicitly. Start with an F7 hull containing the ordered fundamental space and its base assignment. At a successor, apply F7 to the countable set $M_{\alpha }\cup \{M_{\alpha },\alpha \}$ and the fixed parameters; at a countable limit take the union of the earlier increasing elementary chain, which is countable and elementary by the Tarski--Vaught test. F8 performs this recursion, and F9 makes the simultaneous hull choices. Now choose the compatible metrics $d_{\alpha }$ allowed by F1 and F9. For $\alpha <\omega$ put $V_n^{\alpha }=\{\alpha \}$. Inductively suppose the $V_n^{\xi }$ and $H_{\mathrm{\ell }}^{\xi }$ have been constructed for every $\xi <\alpha$, and that the resulting topology below $\alpha$ is locally compact and zero-dimensional.

Fix $\omega \le \alpha <{\omega }_1$. The countable model $M_{\alpha +1}$ contains only countably many sets $S$ satisfying $\mathrm{\mid }S\mathrm{\mid }\le {\mathrm{\aleph }}_0$ and $S\subseteq K(\alpha ,\tilde{\mathcal{T}}\upharpoonright \alpha )$; list them as $⟨S_{\nu }:\nu <\omega ⟩$, repeating every such $S$ infinitely often. This collection is nonempty because it contains $S=\mathrm{\varnothing }$, and that particular $S$ makes the later star implication vacuous.

Recursively suppose $H_{\mathrm{\ell }}^{\alpha }$ is fixed for $\mathrm{\ell }<\nu$ and put $A_{\nu }={\bigcup }_{\mathrm{\ell }<\nu }{H}_{\mathrm{\ell }}^{\alpha }$. This is a finite union of compact sets, hence compact. Every $K\in {\mathrm{\Gamma }}_{\alpha }(S_{\nu },\nu )$ lies in $A_{\nu }\cup (W_{\nu }^{\alpha }\cap \alpha )$: the first $\nu$ ring intersections lie in their $H_{\mathrm{\ell }}^{\alpha }$, while all remaining points lie in the nested $W_{\nu }^{\alpha }$.

The family ${\mathrm{\Gamma }}_{\alpha }(S_{\nu },\nu )$ has a finite Hausdorff $2^{-\nu }$-net $P_{\nu }$ chosen from itself. Indeed, by [F3] choose a finite $\epsilon$-net of the compact metric set $A_{\nu }$, where $2\epsilon <2^{-\nu }$. Partition the remaining tail into the outer ring $R_{\nu }=(W_{\nu }^{\alpha }\setminus W_{\nu +1}^{\alpha })\cap (\alpha +1)$ and the deeper tail $T_{\nu }=W_{\nu +1}^{\alpha }\cap (\alpha +1)$. Record for each $K$ which finitely many $\epsilon$-cells meet $K\cap A_{\nu }$, whether $K$ meets $R_{\nu }$, and whether it meets $T_{\nu }$. There are finitely many records. Two compacta with the same record are within Hausdorff distance at most $2^{-\nu }$: match their points in $A_{\nu }$ through a common recorded cell; points in $R_{\nu }$ are within its internal diameter $2^{-\nu -3}$; and points in $T_{\nu }$ are within $2^{-\nu }$ by the triangle inequality, since every such point has distance at most $2^{-\nu -1}$ from $\alpha$. The two tail-incidence bits ensure that each required matching part is nonempty. Choosing one member for each realized record gives $P_{\nu }$; if ${\mathrm{\Gamma }}_{\alpha }(S_{\nu },\nu )=\mathrm{\varnothing }$, take $P_{\nu }=\mathrm{\varnothing }$.

Let $Q_{\nu }$ be the union of $J\cap (W_{\nu }^{\alpha }\setminus W_{\nu +1}^{\alpha })$ over $J\in {\bigcup }_{\mu \le \nu }{P}_{\mu }$. It is a compact subset of the relatively clopen nonempty $\nu$-th ring. Local compactness and the clopen base below $\alpha$ cover $Q_{\nu }$ by finitely many compact clopen sets lying in that ring; their union is compact clopen. If this union is empty, choose a ring point and one compact clopen neighbourhood of it in the ring. Call the resulting nonempty compact clopen set $H_{\nu }^{\alpha }$; it contains $Q_{\nu }$.

For $J\in P_{\nu }$, membership in ${\mathrm{\Gamma }}_{\alpha }(S_{\nu },\nu )$ already puts its intersections with rings $\mathrm{\ell }<\nu$ inside $H_{\mathrm{\ell }}^{\alpha }$. For $\mathrm{\ell }\ge \nu$, the stage-$\mathrm{\ell }$ definition includes $P_{\nu }$ in ${\bigcup }_{\mu \le \mathrm{\ell }}{P}_{\mu }$, so $H_{\mathrm{\ell }}^{\alpha }$ absorbs the $\mathrm{\ell }$-th ring intersection of $J$. Hence $P_{\nu }\subseteq {\mathrm{\Gamma }}_{\alpha }(S_{\nu },\omega )$.

Put $V_m^{\alpha }=\{\alpha \}\cup {\bigcup }_{\mathrm{\ell }\ge m}{H}_{\mathrm{\ell }}^{\alpha }$. Its maximum is $\alpha$ and its part below $\alpha$ is open. It is compact in the intermediate topology: an intermediate-open cover has a member containing $\alpha$, hence contains some $W_N^{\alpha }\cap (\alpha +1)$ and all $H_{\mathrm{\ell }}^{\alpha }$ for $\mathrm{\ell }\ge N$; only finitely many earlier compact $H_{\mathrm{\ell }}^{\alpha }$ remain. It is therefore intermediate-closed, and [F1] makes the $V_m^{\alpha }$ a clopen local base in the final refinement. The same argument for a final-open cover, now using a contained $V_N^{\alpha }$, proves that $V_m^{\alpha }$ is compact in the final topology.

Whenever $S_{\nu }=S$, steps 4.1 and 6.1 give $\mathrm{\forall }K\in {\mathrm{\Gamma }}_{\alpha }(S,\nu )\mathrm{\exists }J\in {\mathrm{\Gamma }}_{\alpha }(S,\omega )d_{\alpha }(K,J)\le 2^{-\nu }$. Every relevant $S$ occurs infinitely often, so $({\star }_{\alpha })$ holds.

Steps 2.1--7.1 together with step 6.2 perform the successor construction at every countable $\alpha$, while limit stages take the accumulated earlier data. Transfinite induction therefore produces a nice refinement satisfying every $({\star }_{\alpha })$.

Every $V_m^{\alpha }$ is compact by step 6.2, so [F4] gives local compactness. It lies in $\alpha +1$, a countable ordinal, so it is a countable neighbourhood and the space is locally countable. The base is clopen and Hausdorff by [F1], hence zero-dimensional and regular by [F6].

The set $\omega$ remains dense. Inductively, every nonempty relatively open $H_{\mathrm{\ell }}^{\alpha }\subseteq \alpha$ meets the already dense $\omega$, so every basic tail $V_m^{\alpha }$ meets $\omega$; for $\alpha <\omega$ the singleton $V_m^{\alpha }$ itself lies in $\omega$. Thus the refinement is separable.

Every initial segment $\alpha$ is open: if $\xi <\alpha$, then each basic neighbourhood $V_m^{\xi }$ is contained in $\xi +1\subseteq \alpha$. Consequently $\mathcal{U}=\{\alpha :0<\alpha <{\omega }_1\}$ is an open cover of ${\omega }_1$. A countable subfamily has countable supremum $\delta <{\omega }_1$ and its union is contained in $\delta$, so it misses $\delta$. By [F5] the refined space is not Lindelöf.

Combining steps 7.1 and 8.1--9.3 gives all asserted properties. The empty $\mathrm{\Gamma }$ cases were handled in steps 2.1 and 4.1, $\nu =0$ has $A_0=\mathrm{\varnothing }$, the finite ordinals use singleton bases, and F9 records every nonempty simultaneous choice.