If you used an audio interface with multiple inputs, you could record the feed from the mixing board, and also set up a mic to record the crowd noise. Audio interfaces aren't terribly expensive. A Lexicon Alpha interface plugs in USB, and gives you two line-level inputs for the board and an additional XLR mic input. The Alpha is $79.
As far as balancing the sound, this would have to happen during the audio mix in post. If the PA is loud, then the crowd mic will hear it too. It might be a pain to mix in post, but, when you want to capture the sound of a space differently then it actually sounded, then it's going to likely be extra work.
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