fix(badges): return processed data from all badge API functions

CRITICAL FIX (same pattern as event_file fix):
- search__event_badge(): Now returns processed_obj_li instead of unprocessed result_li
- load_ae_obj_id__event_badge(): Returns processed object after IDB save
- load_ae_obj_li__event_badge(): Returns processed list after IDB save
- create_ae_obj__event_badge(): Returns processed object after IDB save
- update_ae_obj__event_badge(): Returns processed object after IDB save

Pattern: All functions now return what was actually saved to IDB (processed data)
This ensures consistency between API return values and IDB cached data.

TEST IMPROVEMENTS:
- Add full_name field to badge mock data (not just full_name_override)
- Ensures mock data matches real API structure

src/lib/ae_events/ae_events__event_badge.ts

@@ -65,6 +65,8 @@ export async function load_ae_obj_id__event_badge({
                     properties_to_save,
                     log_lvl
                 });
+                // Return the processed object (matches what was saved to IDB)
+                ae_promises.load__event_badge_obj = processed_obj_li[0];
             }
         } else {
             console.log('No results returned from API.');
@@ -161,6 +163,8 @@ export async function load_ae_obj_li__event_badge({
                     properties_to_save,
                     log_lvl
                 });
+                // Return the processed list (matches what was saved to IDB)
+                ae_promises.load__event_badge_obj_li = processed_obj_li;
             }
         } else {
             console.log('No results returned from API.');
@@ -245,6 +249,8 @@ export async function create_ae_obj__event_badge({
             properties_to_save,
             log_lvl
         });
+        // Return the processed object (matches what was saved to IDB)
+        return processed_obj_li[0];
     }
 
     return result;
@@ -336,6 +342,8 @@ export async function update_ae_obj__event_badge({
             properties_to_save,
             log_lvl
         });
+        // Return the processed object (matches what was saved to IDB)
+        return processed_obj_li[0];
     }
 
     return result;
@@ -471,8 +479,9 @@ export async function search__event_badge({
                         properties_to_save,
                         log_lvl
                     });
+                    return processed_obj_li; // Return processed data (matches saved IDB data)
                 }
-                return result_li;
+                return result_li; // No cache: return raw API results
             } else {
                 return [];
             }