Noble titles are bestowed by the king, even if it's a high ranking noble family they are still in the palm of the king "on paper", unless they revolt.
I'd argue the king decides the next head of the family when there's a dispute big enough to warrant it.
Though I doubt even the king would have a good time trying to appoint someone without blood ties, that's besides the point though.
PS: Not an expert in royal bullshittery.
I think this is an off paper situation.
Like regardless of who gets to decide stuff like who inherits the Marquis title and who gets to be prime minister
on paper they still have to take stuff like money, power, political alliances, public opinion, etc., into account or their decision won't really "work."
Bernadetta has a legitimate connection to the prime minister, is extremely competent, and there's a precedent of women inheriting noble titles.
Meanwhile it feels like her dad and his faction are about to take a hit from a bad investment: the saintess. Don't remember if she has actual magical powers but it doesn't actually matter, the prime minister's family was
supposed to cultivate her into someone who can garner tons of popular support and win over powerful people and act as a diplomat for the church faction, and they completely failed to do that.
If the situation after they all graduate is something like:
The saintess, who was supposed to bring a ton of benefits to the church faction, is a total dud.
The prime minister's son is also kind of meh.
Bernadetta is extremely competent and is in a position where she
could potentially bring people associated with the prime minister a ton of benefits.
Then that would go a long way towards forcing the prime minister to make Bernadetta his heir.