The counter argument is that the systemic injustices that exist are perpetuated primarily through it being convenient to maintain the status quo.

A capitalists system like this needs a underclass to siphon wealth from.

Thus by fighting the class war you inherently will be tearing down the pillars upholding these injustices. If you get enough Republican voters to have class consciousness the culture war will fall apart on its own because it primarily exists to keep those groups distracted.

class war won’t solve a lot of problems specific to (intersections of) minority groups

I would just disagree with this by pointing out how civil rights leaders like famously MLK Jr actually took the class-first approach and understood that it was economic inequality as a symptom of capitalism, from which "minority groups" and our notion of things like racial identity itself emerged out of. The Populist movement half a century earlier in the US had a similar approach as well, the Jim Crow order was just as much an economic order imposed by the capitalist landowning class to address the threat of the Populists as it was a racial order. Adolph Reed Jr's book The South: Jim Crow and It's Afterlives is a really good explanation of how this worked from a left perspective.

There's a dialogue that continued post-MLK about his support or non-support of the queer community, his widow used his words in support and his daughter used them against. Engaging in some of the academic writing around this specific topic is an interesting way to approach the subject and form your own opinion. Especially in the context of the Christian socialism he advocated, which gets in to cultural issues in a way that's sort of more measured and detached from contemporary culture war hysteria.

Could you provide some examples though?