How much Royal power did Alysanne have over the kingdom?

For a pre-Dance scenario where Daenerys survives childhood but Aemon still marries Jocelyn Baratheon (I'll think of a reason for this later), a fight ensues between Alysanne, who wants to follow absolute primogeniture and make Daenerys the heir to the succession, while Jaehaerys, like in canon, has Aemon as his designated heir.

How much power would Alysanne have, independent of Jaehaerys, to try to change the succession?