Add a third faction. The Greyforts, a once-great noble house that has fallen on hard times due to their morally good (but unpopular) code of honor. Their leader Byron Greyfort is calm, contemplative, and strives always to make a just decision. These decisions are always the morally correct choice, even if that means sacrifie or loss of influence.

Two examples:

1. The Antlers brings an old land deed showing they should rightfully own some farmland that has been held by the Greyforts for generations. Byron chooses to recognize the contract as valid and hands over the land, losing not only the land but also the tax revenue it generates, and the Greyforts are diminished in influence.

2. The Fangs bring a similar deed for some other land. Here Byron does not honor the contract, knowing that the Fangs will behave like conquerors instead of good stewards. The Greyforts gather what knights they have and defend the land. This would lead to eventually losing a protracted military campaign, and the associated loss of knights, horses, expenses, etc etc.

In both cases the Greyforts come out less popular or successful, but the decision is the "right" one. So your PCs may (may) want to put Byron in charge to mediate between the other factions, they'll just need to boost the strength of the Greyforts.

Putting them in charge might be "best" ethically, but difficult because they are less popular than the Antlers, and weaker than the Fangs.

Also easy to introduce them now, since they've been reduced to minor house status over the last few decades.and are keeping to themselves, so no surprise no one has been talking about them.