As said in LD 2x07 Where Pleasant Fountains Lie, Billups is from Hysperia and there is still the technicality that can force him on the throne.
This kind of condition makes me wonder what Hysperia's political status with the Federation is. What stops Billups from e.g claiming asylum with the Federation, especially considering the Hysperians are not a powerful force?
I have three theories:
Hysperia is a Federation colony, so they are able to apply their throne law on Billups and Billups is unable to claim asylum with a political entity he was already born into.
Hysperia is not a Federation member, but has a scare resource (like dilithium) that makes Starfleet want to maintain good relations with Hysperia.
Alternatively, it could be some sort of weird prime directive thing.
My interpretation was not that they can force Billups to return to the planet and perform as king, but that their laws would name him as their king and he is not willing to be negligent in his duties if he is named king because of his personal character.
In the long term, this may mean eventually he will return and become king when it is required of him, but he was trying to delay that and follow his own interests for as long as possible, since the planet currently has a functional and effective ruler: his mother.
This is why he agreed to go through the process of becoming king when he believed his mother had died, again reluctantly, but still choosing to fulfill his duties to respect his family and the needs of the people he would be ruling over, as there is no exploration of alternative heirs or loopholes or workarounds in their legal system.