No, you don't need to resort to a personal visit yet: you can order microfilms of church records. You have to visit your local Mormons to view them, and there's a fee, but it's vastly cheaper than airfare.
I suggest starting with film number 602021 (
https://familysearch.org/films/lookup/products/view/film/602...), which is baptisms, marriages, and burials for Szentgotthárd, 1866 to 1895. This would allow you to confirm the baptisms from the online index, and the marriage records should get you the next generation that you're looking for. Being Catholic records, they're probably at least nominally in Latin, so you don't even need to bring a captive Hungarian along. :-)
For the Czech line, I believe the Czech government has made all sorts of records freely available online, I just don't know any of the specifics.