Could a branch of Buddhism adopt the concept of a holy war? Many actually-existing Buddhists and branches of Buddhism have created justifications for making war. Whether Gurkha mercenaries, Shaolin monks, or Tibetan guerrillas, many notable soldiers have been Buddhist- and still believed their actions morally justified. This extends all the way up to the present Dalai Lama, who has always maintained that Tibetan soldiers were right to defend Tibet against Chinese aggression. So the concept of one of these groups formalizing a theory holding that defending the faith by attacking threatening outside groups is morally required rather than merely morally justifiable seems to me to be only a single step, albeit a large one.
Would this lead to a wider spread of Buddhism? I would guess no. Buddhist teachings don't seem to have ever coincided with the conditions that made holy war doctrines effective for Islam or, to a lesser extent, Christianity. For proselyting by the sword to be effective you had better win, a lot. Christianity's limited success with crusades followed after acquiring the sponsorship of the Roman Empire, and Islam's greater success with Jihad required as a precondition for success the near-total disintegration of the two largest Empires in the regions surrounding them, with later Islamic efforts at expansion through conquest, whether in Europe, Ethiopia, or Asia much less successful than expansion through trade and conversion. Buddhism seems to me to be more likely to suffer by convincing potential converts that the faith is dangerous and should be opposed than to triumph by the sword.
I think determining the wider theological impact would be very hard, simply because of how much the circumstances around the adoption of this new doctrine would matter. One point I will make- a lot of Buddhism's impact on modern western thought and spirituality is highly filtered through Japanese Zen concepts, simply because that was the most accessible version when Westerners really started to take an interest in the faith. A doctrine of holy war originating from, say Nepal, might easily be dismissed as an unfortunate local folk superstition grafted on to the true religion, much as Tibetan and Nepalese Buddhism's focus on gods and spirits is still frequently dismissed as not "really" Buddhist.
Would this lead to a sense of Buddhist unity? I'm skeptical. Of course, much depends on when and how this doctrine comes about, but most likely scenarios I can think of (say, Buddhist preachers in Imperial Japan in the 1920s and 1930s being even more explicit than OTL about the connection between violence as a tool of imposing social order and Buddhism) seem more likely to alienate any Buddhists who don't belong to the specific branch and specific nationality at issue- splintering the faith further rather than bringing it together.
