feat: keep rust click-through cache synced with UI toggle

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

src/hooks/useLiveOverlayWiring.ts

@@ -4,6 +4,7 @@ import {
   applyAlwaysOnTop,
   applyClickThrough,
   applyOpacity,
+  syncClickThroughCache,
 } from "../lib/overlay";
 
 /** Pushes Zustand changes to the overlay window in real time. No-op when overlay is closed. */
@@ -30,5 +31,6 @@ export function useLiveOverlayWiring(): void {
     if (lastAppliedClickThrough.current === clickThrough) return;
     lastAppliedClickThrough.current = clickThrough;
     void applyClickThrough(clickThrough);
+    void syncClickThroughCache(clickThrough);
   }, [clickThrough, isOpen]);
 }