Unfortunately, if you've already tried recon and had no success, that's the end of it.
Prequalifications are "firm offers of credit" but are not set in stone. There is always a paragraph of tiny print on there somewhere that says, more or less, "if, under more intense scrutiny than we got from our soft-pull glance at your credit file, you fail to meet our unpublished protected trade-secret formula for what we can accept, we will refuse to offer you a card even though we sent you this pre-qualified letter and coerced you into trying. Please enjoy the HP(s) we leave you as a souvenir."
In your case, their prequalification computer program did not check your inquiries (most of the ones out there don't, actually) and so it spit out offers for you.
Their actual approval computer program did check your inquiries (all of them do, just depends on the lender's policy what weight they put on them), and your file did not pass.
Sorry about your frustration. Pre-qualified offers can be very tempting, especially for those who are looking to build or rebuild credit, but ultimately they do not obligate the lender to offer a card, and can lead to disappointment (and extra HPs). Anyone considering applying for anything should be aware of the generally observed habits of the lender in question with regard to score ranges for acceptance (although this is very fluid across the board), sensitivity to inquiries, and tolerance for derogatory entries.
