Update the OwnerDraw flag when BackgroundImage changes

src/System.Windows.Forms/PublicAPI.Unshipped.txt

@@ -1,3 +1,4 @@
+override System.Windows.Forms.Button.BackgroundImage.set -> void
 override System.Windows.Forms.ButtonBase.OnBackColorChanged(System.EventArgs! e) -> void
 override System.Windows.Forms.ButtonBase.OnForeColorChanged(System.EventArgs! e) -> void
 static System.Windows.Forms.Application.SetColorMode(System.Windows.Forms.SystemColorMode systemColorMode) -> void

src/System.Windows.Forms/System/Windows/Forms/Controls/Buttons/Button.cs

@@ -148,6 +148,18 @@ public virtual DialogResult DialogResult
         }
     }
 
+    public override Image? BackgroundImage
+    {
+        set
+        {
+            base.BackgroundImage = value;
+
+            // BackgroundImage changes may affect rendering logic,
+            // so we manually update the OwnerDraw flag to ensure correct visual behavior.
+            UpdateOwnerDraw();
+        }
+    }
+
     /// <summary>
     ///  Defines, whether the control is owner-drawn. Based on this,
     ///  the UserPaint flags get set, which in turn makes it later
@@ -161,7 +173,6 @@ private protected override bool OwnerDraw
         get
         {
             if (Application.IsDarkModeEnabled
-
                 // The SystemRenderer cannot render images. So, we flip to our
                 // own DarkMode renderer, if we need to render images, except if...
                 && Image is null