I don't think the Monarch name is a good one. Unless you want Betsy stuck in Otherworld for the foreseeable future!(Out of sight, out of mind)
Beside Avalon is being passed around too much! First if was Arthur's and to my knowledge he's still the legitimate king of Avalon. Now it's in Jamie's hands thanks to Apocalypse, who doesn't really have a reason to give it up. It's can't now go to Betsy, who just became CB after wanting to always have the role. It's just too much mess for the character. Let her land somewhere for a while. Since the split from Psylocke, it doesn't really feel like there has been an attempt to either build a new foundation for the character or pick up from where she left off before she merged with Kwannon.