Thank you - quite apparent as the flight is almost free.

Do you know if I would (could) be offered an upgrade on the Swiss website if I book this (third-party website, LX force me onto a direct codeshare).

I doubt it. I suspect LX would rather put you in the cargo hold when travelling on an E class fare.

If you want to upgrade using miles (also doubtful on this cheap fare) suggest you look at the M&M forums although I think you already know the answer to your question.