Er, could you not just do it on the QT? I'm sure once you get a few students the referrals will come in. Many English teachers do this: start giving occassional 'conversation' classes - cash in hand - and before they know it - if they're any good - they have a small business on their hands.

When it comes to AHV, I understand you can retrospectively let them know what you made and how much is owed them. Then the tax people will be happy to cooperate if you're transparent. The only problem I see is your permit. Do you have a significant (Swiss) other who may help?

Be wary of free information on teh interwebs