Fix: Harden V3 search logic and restore specialized business mapping

- API: Updated `search_ae_obj_v3` to correctly serialize complex URL parameters (JSON).
- Events: Restored "sacred" business logic for Event Badge and Session searches using `ft_qry` and `lk_qry`.
- PWA: Fixed manifest path in `app.html` to resolve 404 errors.
- Documentation: Updated `GEMINI.md` and `TODO.md` with recent search hardening accomplishments.

src/app.html

@@ -4,7 +4,7 @@
         <meta charset="utf-8" />
 
         <link rel="icon" href="%sveltekit.assets%/favicon.png" />
-        <link rel="manifest" href="/manifest.webmanifest" />
+        <link rel="manifest" href="/manifest.json" />
 
         <meta name="viewport" content="width=device-width, initial-scale=1.0" />