Re-reading my post, I never addressed the original concern: Why is 嫌い a na-adjective?
Originally, the verb 嫌う was nominalized using the conjunctive form to create the noun 嫌い = hatred. Over time it came to be used as an adjective and, stemming from a noun, was treated as a na-adjective (which conjugate like nouns anyway).
嫌い is by no means the only such adjective, but it is by far the most common.
A few other examples:
場違いな発言 (originates from 違う) = out-of-place remark
不似合いな夫婦 (originates from 合う) = ill-matched couple
大食いな人 (originates from 食う) = glutton
幸いな結末 (originates from 幸ふ) = happy ending
It might be easier to just memorize the handful of "exceptions" when you come across them, rather than think through their etymology, but there is some historical reasoning behind it all.
Nice explanation. This makes me wonder if it is possible to make a general rule that nominalized う-verbs become な-adjectives when used to modify a noun. Though I expect there would be exceptions.