Medieval wars were risky and costly - villages had to supply armies with essential food and working population, and defeat meant wasted resources and dead villagers. Haven combat, on the other hand, is trivial. Even if your character dies, it can inherit most of the ancestor's LP, and the village's economy is left unharmed. Armors don't have to be repaired after battle, weapons never decay, and mutilated members are always heal.
My suggestion is that combat is turned into a riskier gamble, with much more stuff at stake. When a character is killed, its village could lose a large amount of authority. Upon reaching zero, the village idol is automatically destroyed. Or maybe all other villagers would lose a % of their stats. It's just a general idea. Aside from the collective risk, the personal risk of losing LP could be enhanced. If LP inheritance is removed, the stakes are higher when you enter combat, even with alt abuse.
To limit use of alt villages to circumvent this consequence, characters with UA and MC above a certain value would regularly drain X authority from the idol it's oathed to, representing the village's support he receives. That means a good fighter would likely to be oathed to a major village to avoid complete auth depletion, and that still keeps the risk of defeat high enough to not disregard.
The winning players should have better rewards as well. Maybe whoever gives the final blow to a village's auth should receive LS of it's idol? Skulls could give much more LP than they do now, or maybe even yield you additional stat points. You could make a forum poll to decide this.
This would, in my mind at least, give a more concrete political weight to pvp combat and also make combat more rare and more meaningful. With the right tweaks it could direct players to form large villages - not a series of villages around the world, but a huge city to support their fighters' upkeep and with enough authority to resist defeat after their fighters are killed in combat.