I think the thing about 大空 is that it’s not a compound kanji, but a compound word. It is a slightly difficult distinction to get. I guess 大い isn’t taught as a word because no-one really uses it any more except as the adverb 大いに (ha ha ha like that ever stops WK, but joking aside it’s so archaic even my IME won’t produce it), but 大空 is a contraction of 大い空. It’s two words that have been compounded into one, rather than one word that requires two kanji. So it uses the readings of the constituent words, which is kun’yomi in both cases.
That doesn’t really help with how you know the difference, but hopefully it helps with understanding why there are differences.