Indeed, the onGuiHandler is never called for the MouseDown/MouseUp events if the multi colown tree view is used inside an IMGUI container, which is why the the UI cannot be interacted with. If the multi column tree view is not used inside an IMGUI container, the issue does not occur.

The root cause of this issue is deep within the framework and will not be solved at this point, however we can offer a workaround: you may set the height of the IMGUIContainer like in this modified snippet taken from MultiColumnWindow.cs

private void CreateGUI()
{
var root = rootVisualElement;
var container = new IMGUIContainer(() =>
{
InitIfNeeded();

// SearchBar (toolbarRect);
// BottomToolBar (bottomToolbarRect);
DoTreeView(multiColumnTreeViewRect);
});

// Make sure the height of container is not zero so that the IMGUI
elements receive mouse events
container.style.height = new StyleLength(new Length(100, LengthUnit.Percent));
root.Add(container);
}