That's excellent news!
As for payment, you need to check what your bank charges are for an international payment. Here in the UK it's very expensive for small payments. What I have done which might be useful for you, is to check what an insured & signed for delivery postal item would cost, which here was a great deal less than the bank transfer. Then get the nearest greater sum of Swiss Francs cash from your local forex shop, and post it. Worked for me a number of times!
You need to ask your contact at the Civivl Registry if they have the parents familienschein, but the civil records might peter out about that time. The only next source will be the Muhlethurnen church records, which are on microfilm (8 of them!) that can be ordered to view at the LDS FHC nearest to you. Look at the familysearch website library catalog for full details inc film numbers and they contain records from 1549. I can't work out if they are a transcription or original, so could be in old German script and may be near unreadable to our eyes.
Paul