For capitalism to be part of a culture war, it would have to take a side. Capitalism doesn't care about any culture beyond its ability to assimilate, commodify and market the superficial features of that culture as a product. Capitalism has even done it to communism - look at how much Che Guevara merch there is out there.

Capitalism does "care" about culture that is needed to sustain capitalism. E.g. maintaining coercive structures upholding property claims, promulgating ideologies that support capitalism and supressing ones that don't. This happens via e.g. campaign funding, public relations, think tanks, wars etc.