feat(chat-overlay): add ChatBox config UI + schema persistence

- Elements panel config UI for ChatBox source (font size slider,
  font/member/super-chat/background color inputs, max messages slider,
  timestamp/badge toggles)
- Schema migration: 8 new ChatBox columns on Source table (ChatFontSize,
  ChatFontColor, ChatBackgroundColor, ChatMaxMessages, ChatShowTimestamps,
  ChatShowBadges, ChatMemberColor, ChatSuperChatColor)
- Load/save: Source SELECT and INSERT now include all ChatBox fields
- IsElementsPanelEmpty excludes ChatBox so hint doesn't cover config
- ChatBoxRenderer.cs created: WPF FormattedText + RenderTargetBitmap
  renderer for on-stream chat overlay
- Slider bindings set to TwoWay for live propagation

ViewModels/MainViewModel.cs

@@ -126,6 +126,7 @@ public class MainViewModel : ViewModelBase
     private readonly IFullScreenDetector _fullScreenDetector;
     private readonly ScreenCaptureManager _screenCaptureManager;
     private readonly ScreenCaptureSourceFactory _screenCaptureFactory;
+    private readonly ChatBoxRenderer _chatBoxRenderer;
     private int? _lastForegroundFullScreenMonitor;
     private CancellationTokenSource? _deactivateCts;
     private ImageSource? _backdropImage;
@@ -270,7 +271,7 @@ public class MainViewModel : ViewModelBase
     /// <summary>True when the Elements panel should show the "No configurable elements" hint.</summary>
     public bool IsElementsPanelEmpty => SelectedElement is { } el
         && !el.IsDraggable && !el.IsWebcam
-        && el is not Source { Type: SourceType.Countdown or SourceType.WebSource };
+        && el is not Source { Type: SourceType.Countdown or SourceType.WebSource or SourceType.ChatBox };
 
     /// <summary>
     /// The reason the webcam feed is down (device in use, offline, locked, no frames),
@@ -1267,6 +1268,8 @@ public class MainViewModel : ViewModelBase
         _screenCaptureManager.CaptureFailed += (key, message) =>
             AppLog.Write($"ScreenCaptureManager: capture '{key}' failed: {message}");
 
+        _chatBoxRenderer = new ChatBoxRenderer();
+
         _rtmpUrlProvider = () => _reusableStreamUrl; // TASK 5: the reusable stream's ingest URL
         _framePump = new FramePump(
             sceneProvider: () => ActiveScene,
@@ -1761,6 +1764,7 @@ public class MainViewModel : ViewModelBase
             SourceType.TextOverlay => "Text",
             SourceType.Countdown => "Countdown",
             SourceType.WebSource => "Web",
+            SourceType.ChatBox => "Chat",
             _ => "Source",
         };
 
@@ -2414,11 +2418,33 @@ public class MainViewModel : ViewModelBase
         {
             WebcamSceneConfig webcam => _cameraManager.GetLatestFrame(webcam.WebcamId),
             Source { IsLiveCapture: true, CaptureKey: not null } live => _screenCaptureManager.GetLatestFrame(live.CaptureKey),
+            Source { Type: SourceType.ChatBox } chatBox => RenderChatBox(chatBox),
             Source { AssetId: not null } image => StaticPixelCache.Get(image.AssetId),
             _ => null,
         };
     }
 
+    private VideoFrame? RenderChatBox(Source chatBox)
+    {
+        if (ChatMessages.Count == 0) return null;
+
+        var width = (int)Math.Max(1, chatBox.Width);
+        var height = (int)Math.Max(1, chatBox.Height);
+
+        return _chatBoxRenderer.Render(
+            ChatMessages,
+            width,
+            height,
+            chatBox.ChatFontSize,
+            chatBox.ChatFontColor,
+            chatBox.ChatBackgroundColor,
+            chatBox.ChatMaxMessages,
+            chatBox.ChatShowTimestamps,
+            chatBox.ChatShowBadges,
+            chatBox.ChatMemberColor,
+            chatBox.ChatSuperChatColor);
+    }
+
     // The tier's crop rect over the 1920x1080 master, integer-aligned (the VM's
     // OutputRect* are doubles — the vertical 607.5 half-pixel crop rounds to 608).
     private CompositorOptions BuildCompositorOptions()