> But why is this a top-3 culture war issue?
It's not. Without suggesting these are the top 3, Abortion, gun control, and immigration are, just to pick 3 obvious ones, a far more central culture war issues than pronouns. (If you aggregated “LGBT rights” as a whole, you might get into the neighborhood of a top-3 issue.)
> For a bunch of combatants who are overwhelmingly not trans and shouldn't really care?
Unless, of course, you are a combatant on the other side, I can't imagine why you'd think that not-trans people shouldn't really care.
> We need to think of a way to rise above without throwing anyone under the bus.
How isn't having the decency to respect people's preferences in the pronouns you use to refer to them (and, though it's a separate but often concurrent issue, the gender you ascribe to them) exactly that?