I think you're making this more difficult than it needs to be.
What you need to do is sit the girl down and tell her that regardless of her relationship with you and with your VPR, NO ONE SISTER CAN GET A PNM A BID BY HERSELF.
Tell her that no matter what she thinks she heard, even if she heard right (which we know she didn't and you need to document, just in case), that doesn't mean the rest of the chapter will vote to give her a bid.
Remind her that one vote is one vote is one vote. That's all. Majority rules. The VPR's vote counts for no more than any other sister's vote. Then tell her that voting is a private process that is based on sisterly trust, and that a chapter must always respect the majority decision of its sisters, regardless of individual or minority opinions.
Please don't fib to her about grades or legacies, etc. etc. etc. If/when she does not recieve a bid, tell her that the majority of the chapter just didn't feel like it was a good fit--for her or for the chapter. That's utterly legitimate, and it's the truth.
At the same time, even though you say that you and your friends don't see her in your chapter, you never know--this PNM could go through structured recruitment, and a majority of your sisters could decide that she fits in just great....just as you think this PNM shouldn't count her AGD chickens before they are hatched, neither should you!
Best of luck with this one!