refactor(badges): fix text search and standardize field mapping

- Renamed 'default_qry_string' to 'default_qry_str' across the Badge interface, Dexie schema, and API logic to resolve 400 'Unauthorized search field' errors.
- Synchronized local Fast Path filtering with the correct database field names.
- Hardened the reactive search pattern in '+page.svelte' to ensure end-to-sync consistency between local and background results.

src/lib/ae_events/db_events.ts

@@ -159,7 +159,7 @@ export interface Badge {
 
     // data_json?: null|string;
 
-    default_qry_string?: null | string;
+    default_qry_str?: null | string;
 
     alert?: null | boolean;
 

src/routes/events/[event_id]/(badges)/badges/+page.svelte

@@ -156,7 +156,7 @@
                                 const family_name = (badge.family_name ?? '').toLowerCase();
                                 const full_name = `${given_name} ${family_name}`.toLowerCase();
                                 const email = (badge.email ?? '').toLowerCase();
-                                const qry_string = (badge.default_qry_string ?? '').toLowerCase();
+                                const qry_string = (badge.default_qry_str ?? '').toLowerCase();
                                 
                                 const match = full_name.includes(qry_str) || 
                                               given_name.includes(qry_str) || 
@@ -207,10 +207,6 @@
             } catch (e) {
                 if (log_lvl) console.warn('Badge Fast Path failed.', e);
             }
-        } else {
-            untrack(() => {
-                event_badge_id_li = [];
-            });
         }
 
         // 3. REVALIDATE: API Request