How do I appropriately decline an invitation to an appointment lecture ("Probevorlesung") without burning bridges?
You should communicate clearly (and swiftly) that you are declining the invitation because you are accepting the offer of a professorship at another university. In fact you should not just 'decline the invitation', but completely withdraw your application to your hometown university.
I would not be worrying too much about burning bridges, most likely the view of the hiring committee at your hometown university will be more like 'Shoot! We have lost an excellent candidate to university X. We have to be more attractive/move more quickly/offer better working conditions next time'.
Finally, let me say that this seems like a true luxury problem! Being offered a professorship 'a little further from where you live' and being invited for a Probevorlesung at your hometown university is something that most aspiring researchers would only dream of.