One thing to know is that the pacifistic aspect of Buddhism applies to those who have individually vowed to honor the precepts. You can be a fan of Buddhism or even be a donor to Buddhist causes, but unless you have taken those precepts, you aren't actually breaking any rules. Some lay people take a version of the precepts but usually it is monks who would do so. A Medieval Japanese samurai who has not taken the vows would be unlikely to feel any sort of cognitive dissonance.

That said, there were still many monks throughout Japanese history that took the precepts yet supported or were involved with combat.

The Sohei Monks for example had actually armies and were a force to be reckoned with. How did these ordained Buddhist Monks justify their behavior? The general philosophical justification is basically that a violation of secular law is indirectly a violation of religious law and also they believed they were living in a time of religious decline that necessitated drastic measures. Mikael Adolphson makes the point in his paper "Discourses on Religious Violence in Premodern Japan":

This brings us to the second type of ideological justification, known as the “interdependence between Imperial Law and Buddhist Law” (ōbō buppō sōe).24 The concept first appears in a tenth-century document, which explains that a violation of the Imperial Law was also a crime against the Buddhist Dharma. A more explicit case emphasizing their interdependence emerges in 1053, in an appeal filed by an estate manager from Tōdaiji, where the complainant compares the laws of the state and those of Buddhism to the two wings of a bird and to the two wheels of a cart.25 The interdependence of these spheres was used to justify violence in times of a perceived decline, when Buddhism could be seen as the final line of defense to protect the imperial state.

A point he also makes in the paper is that the monks believed that they had divine justification because the Kami or indigenous nature spirits were on their side since they were "good monks".

To look at an Actual Zen Buddhist monk we can look at the writings of Takuan:

Stated in terms of your swordsmanship, you should not think about techniques for striking with a sword. Forget all striking techniques and strike. Cut the other person, but do not dwell on the other person. Both self and other are emptiness. The striking sword is also emptiness. But, do not let your mind be restricted by emptiness.

He is saying that from the view of Awakening (Enlightenment), all things are One. There is no self and other, so nobody is harming anyone. This is a time honored rhetorical justification but a flimsy one.