I honestly don't know what this legacy's attitude is like. I don't know if the other chapter is making something out of nothing because she is a legacy, (which is rare at my school), or if she is, as you described, a brat.
Either way, even if this other chapter's request that you be allowed to give her a bid before recruitment starts is granted, that doesn't mean your chapter HAS to give her a bid before recruitment starts.
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