Here's another thought regarding definitions. Green's novel has all of the eight elements that I have identified. The hero dies. The heroine has a form of cancer that is very likely to kill her long before she reaches middle age.
Yet the story ends with the words of the marriage vow: "I do" written by the heroine, our first-person narrator, in response to a posthumous question (delivered via a letter) by the hero. He has said earlier, "I fear oblivion." But in our hands is the book, in which his beloved immortalizes him. No oblivion for him.
I think this qualifies as an HEA, given the constraints of illness that the hero and heroine operate within. The usual meaning of happily ever after implies an expanse of time that is unbounded. Over and over again, The Fault in Our Stars insists on the limited time that we all have, not just those of us with life-threatening illness. So the "ever after" in this HEA has been achieved, I think.
RNA would count it within their definition, I suspect. RWA's definition would also count it. The more restrictive "courtship and betrothal of one or more protagonists?" Yep, I think so.
Quite aside from any marketing, branding, or other marketplace issues, it seems to me that, fomally, this is a romance novel.
What do you think, folks? At the very least, this would be a wonderful topic for a PCA proposal! (Tick-tock: only a dozen days left to submit!)