NOTE: I had a philosopher friend, who did his dissertation on the Nicomachean Ethics, who insisted the best rendering of eudaimonia was "flourishing" -- but that strikes me as even more problematic than "happy."
Fully agree. "Flourishing" seems ... wrong? Merriam-Webster defines "flourishing" as
"marked by vigorous and healthy growth; very active and successful."
Your "happy well-being" is better, for sure. I lean more in the direction of "satisfied well-being" or "contented well-being." To be even more literal, I might suggest "to be in good spirits," but that might be taking the linguistic pun too far.
It was impressed upon me (by Peripatetics) in college that "flourishing" was the only acceptable definition of εὐδαιμονία. One of the professors of the department described Aristotle as "the smartest person who ever lived." The department head was particularly found of Alasdair MacIntyre and assigned us his writings.
I tend to take the word as meaning "good-spirited-ness", in which case, "happiness" is a reasonable translation.