feat(#42): FIX projects can't add scenes (studio + admin)

Template copy now carries choose_mode from the content project → studio store gets
chooseMode; AddSceneMenu returns null for FIX/MusicVisualizer. Admin ProjectScenes
hides '+ صحنهٔ جدید' (shows an 'scenes defined in AE' note) for fixed modes. Verified
choose_mode=FIX flows end-to-end. (Visible admin nav link added earlier.)

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

services/studio/FlatRender.StudioSvc/Application/Services/StudioService.cs

@@ -193,6 +193,14 @@ public class StudioService(StudioDbContext db)
             FROM content.shared_colors sc
             WHERE sc.project_id = {1};",
             savedProjectId, originalProjectId);
+
+        // 5. carry the template's render mode (FIX/FLEXIBLE/…) so the studio knows whether
+        //    adding scenes is allowed, plus a few useful project defaults.
+        await db.Database.ExecuteSqlRawAsync(@"
+            UPDATE studio.saved_projects sp
+            SET choose_mode = COALESCE((SELECT p.choose_mode::text FROM content.projects p WHERE p.id = {1}), sp.choose_mode)
+            WHERE sp.id = {0};",
+            savedProjectId, originalProjectId);
     }
 
     public async Task<SavedProjectFullResponse> UpdateProjectAsync(