I'm now pretty sure this isn't possible using only Plotly.NET as all it does is generate the HTML for the iframe as Joe pointed out.
The way the Python version works is by communicating with the Jupyter engine and send messages around (I haven't fully understood these layers though)
The way to do this I believe is:
With a bit of luck I might be able to find the code to serialize that data/layout in Plotly.NET too
(btw, I'm a big fan of LINQPad @joe and I've hacked around in it quite a lot)