@goodhunter
While you are correct, there is just the 3/4 of them at the moment. MC clearly has final say, but there isn't much of a power hierarchy established beyond that yet besides maybe Eve being next highest. And since each is completely different in terms of their area of expertise and personal skills, they should only rarely conflict with each other anyways. And one may say that them being allowed this amount of freedom is a positive because it shows how capable they are in their own field where they do not need constant observation in order to do what is needed of them.
That being said, this is more of a lack of manpower issue than an "I want to go out on an adventure with MC" kind of deal. If they had more man power, then there is no reason for why both of them couldn't go, as both do not really have the necessary skills for holding governing/political positions nor leadership skills to ensure shit gets done right when the boss is away. At best I would assume they could manage to keep the status quo going for a while, but not forever as that is not their area of skill.
He really should leave behind Eve, as she is suited for this general management business, but as she is mostly a non-combatant besides for directly working with MC, she can't lead a defense from the front lines by herself.
This duel is bad for moral though. Because there should by now be factions established, and when one of them loses, then that faction loses power. The best case is that they tie and MC pushes rhis as a combat demonstration and not a duel for who gets to go on a trip with Mom and Dad.