OP you’ve made the mistake of presuming the world at large knows what DNI stands for.
I’m also in the “stop butchering my language” camp, however I see you don’t mean using Latine or Latinx as a replacement for Latino and Latina in general. You are asking whether non-binary people prefer Latinx or Latine in their personal ID documents. Which is one of the few cases I think using the term is justified.
But since I’m not part of your demographic in question I don’t know if my answer is valid. I guess Latine is at least easier to pronounce than Latinx so there ya go.