I don’t think there is a solution. But there is a workaround:
Define your own Text Visualizer and put appropriate DebuggerVisualizer attribute before the attribute of your JsonVisualizer. The result will be that string will be readable by default and Json Visualizer can be chosen. A window with a multi-line textbox is not too much work.
It is probably not even necessary to write visualizer. It should be possible to use internal one but I don’t know its name (Which class is used for “Text Visualizer”?).