I suspect one of the aspects of Hunter v. Fighter is the value to the story. I saw several members of my group trying to push up Fighter in order to better face off against their perceived opponent: combat. However, I wanted to illustrate the larger scope of MG and importance of facing a variety of challenges effectively to serve their community of mice.
I believe one of the reasons the players perceived combat as the largest winning factor is related to playing D&D encounters immediately before our MG sessions.
A mouse which has concentrated on fighting at the expense of hunting has likely made a name for himself as a violent soldier. Perhaps a guardmouse like that was viable during the Weasel War, but now that sort of hot-headed mouse presents a strange lack of Nature (Mouse) all around. The reputation may include a mouse which thinks first of attacking a problem–attacking a mouse believed to be guilty, attacking mice believed to be a threat, attacking mice that have different opinions, etc.
A mouse which has placed a higher emphasis on hunting and allowed fighting to fall away from the priority skills may yet illustrate a mouse with a strange Nature (Mouse) whne one considers that mice are primarily herbivores–they typically do not need to hunt animals for food. Yet, the ability to protect a community from predators may appear more universally useful and more appealling to other mice than fighting.
This will be reflected also in the sort of campaign setting you describe.
I forced my group to make Weather Watcher tests, Administrator tests, and Survivalist tests knowing that few mice in their patrol had those skills. I watched them seeking to use Fighter in conflicts that were not appropriate to the skill. I more frequently heard them present goals which seemed, ‘My way or the highway.’ Their BIGs frequently revolved around fighting. The patrol had trouble facing obstacles which did not cater to an aggressive, violent attitude.
It was leading to their patrol gaining a reputation among the territory and the Guard of violence, selfishness, and insensitivity. We unfortunately ended before a winter session in which Gwendolyn was intending to place several of their patrol on probation for choices made during patrols that were not within their jurisdiction to make. They had disregarded settlement issues when attempting to mediate.
I had planned for about one Fight conflict per 6 sessions. This left the group woefully unprepared for facing other conflicts and complex tests during other sessions.
You can choose to make a campaign that encourages and empowers Fighter over Hunter. I think that the value comes not from mechanical usefullness, but instead from story value. In my story the mice which focused on Fighter were becoming a serious liability to the Guard; Gwendolyn was unhappy with the damage control from their conduct; the patrol was less capable of serving the community effectively. In your story, you may find that Fighter is far more valuable as a skill and provide a campaign that rewards the investment; those mice are better prepared to serve the Guard; the patrol is better served for their presence.
Mechanically, you are seeing correctly that Fighter can be applied to both Fight and Fight Animal while Hunter can be applied only to Fight Animal and Hunter tests. However, that presents only half of the game. You must now decide what is more valuable in the campaign you want to present. Bring the campaign to life through compelling reasons that Fighter or Hunter has more value than the other. You can also create compelling reasons why Baker, Cook, or Harvester are key skills for the campaign.
Which ever direction you go, let the players understand the setting enough to determine how much their own mice values the skills and desires to fulfill those skills.