This does not apply only to 大, but my approach when I learn a word is to associate its meaning with the sounds of the pronunciation, not the kanji themselves, and it works great. I know the word だいじ means “important”, I know the word たいかい means “tournament”. When I hear them I recognize them like that. When I see the kanji, I use that as a shortcut to get to the word that’s already memorized as kana in my head. So if I see 大会, I try the different readings in my head first and see if I find a match. I don’t recognize だいかい so I try たいかい. It matches the word I know, so I’ll pick that reading.
In my experience, that’s the easiest way not to mess up the different readings.Instead of seeing the characters and trying to guess the readings, I memorize the word in kana and use the kanji as a tool to get to the kana that I already know. Not sure how much that helps you, but I hope it does.
After learning a lot of kanji, you kind of get a feeling for what readings are the most common, and can guess the reading of a new kanji word you haven’t learned before, but that will take a lot of learning until your brain learns to recognize these patterns.