What about an instanced system? It would kind of be like FFXIII where you're on the map, you run into the enemy and you enter battle.

For FFXIV, you would see the HNM (let's use a Wyrm as an example) like any other mob in the area. Then, once you do something to gain enmity on it, your group is transferred to an Instance where you can fight the mob and special things can happen in the battlefield.

Then, once the HNM is defeated, it would disappear from the world until it's respawn timer is up. Any group in an instance against the HNM would be able to finish the fight and obtain drops.

In this way, it wouldn't matter who "claims" it first, because all groups would have an amount of time to enter their own instance and fight the HNM up until it's defeated.