Similar to T74847 the wikibase settings are hardcoded into the family files. There is action=query&meta=wikibase which would allow to make dynamically if the API plays along. In T74847 a very nasty problem surfaced which made it at least very hard to determine the Site object parameter (basically which family and code it uses) based on the result (see T85153).
So part of this task is also to determine if this API call has the same problem so other wikibase installations in the wild should be queried (or more exact: installations which use another wikibase). At least from the result of the English Wikipedia it should be easy to get a Site object because it's possible to construct the URL similar to the interwiki map URL for which we already have an implementation.