I could totally see the Federation getting involved in a war with the best intentions, believing that they need to act to prevent a powerful, implacable faction from gaining the upper hand and tipping the balance of power across many star systems & potentially exterminating/enslaving multiple less advanced civilizations, say along a Federation border.
And the whole thing starts going sideways, and the Feds won’t commit fully to the effort but a dedicated faction of true believers keeps it going on a shoestring.