Why couldn't you write alternating chapters?

IMHO, it creates quite some suspense to see either side rush toward the clash (in my story's case, literally).

And when both are vital for the story it'd be justified as well... because both are the halves that make the story whole.

So I'd go at it.

I'll do so with my own story, too, as the – especially emotional – dynamics between my leading couple make a huge part of it... being the blood that rushes through its veins and (hopefully!) makes it come alive.