I used to work for the US Census Bureau and, according to their definitions, you’d check Asian for race and Hispanic for ethnicity. Answering that way doesn’t “go against the spirit” of the question; it’s honestly answering exactly what the question is asking. If Hispanic was classified as a race (and there was talk then that eventually the Census Bureau will classify it that way), that would be different. But it’s not.
I think your plan to check both boxes and use your essays to give insight into your experiences is a good one. Good luck.