Here's an idea my linkshell leader came up with and personally I think this is the best idea yet.

Make roaming HNMs pop and roam but have depop timer. Also, and this is most important. When you attack it your group goes into an instance. Like in the single player FFs when you go into battle.
This way you could make a HNM pop once every 24 hours, stay up for 15 min and any group that attacks it gets to fight it. You would be able to attempt it once per pop. Assuming there are other HNMs like this you'd have to choose which you want to attempt. It would prevent people from botting or monopolizing them and it would bring more life to the environment.