I want to either disable the menu of a specific web part on my SP-site, or instead disable the "Edit Web Part" function.
This is not for the whole page, but for that specific web part.
Is the WebPart a custom WebPart implemented by you? If yes, you could probably override the AllowEdit property and return false.
You can put the Web Part into the page outside any Web Part Zones, which prevents anyone from editing it. You would do this by editing the page in SharePoint Designer.