We do use the term state capital.
"Oklahoma City is the state capital of Oklahoma."
For Japan. I'd say something like:
"Nagano City is the capital of Nagano Prefecture."
"Prefectural capital" is probably an option but it sounds 'forced.' Could you just drop the 'regional' part and still have it be understood that you are talking about a smaller area. In this way, it's clear that Lombardy is a region/area and Milan is its capital.
"Milan is the capital of Lombardy."
It sounds fine to me this way. 'regional capital' might cause me to wonder what that is. (But of course, not to seem ignorant I wouldn't interupt you. I'd just pretend I understand
)