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When I got my First Cap1 Credit Card back in 2009, I only had SSN (through School), I wasn't a US Citizen, nor did I have Residency. I was an International Student, wasn't allowed to work either at the time! For my Income, I would put $24k/year as my Father would send me money.
They did ask me to provide them the proof that I indeed was documented, and that's all, they approved me!
I know a bunch of F1 students (which was also my status when I got my first Cap 1 card) who have Cap 1 cards* so I am not inclined to think OP's troubles are (solely if at all) related to OP's residency/citizenship. Nothing in their terms AFAIK necessitates citizenship (or permanent residence), just that one lives (has an address) in the US and has verifiable income. Like the poster above, I also had to verify identity and income and I did in fact at that time send them a copy of the relevant parts of my clearly foreign passport (and my US visa) among other things and everything was ok.
*With (legal) SSNs though.
Edit: Going through the application on their website, it does seem that the app requires an SSN, the ITIN only being mentioned in the terms once, but it doesn't really say you can use an ITIN to apply, so the problem could be that OP does not have an SSN. The ITIN is in the same general format, but has specific numbering requirements, so I wouldn't think they're interpreting your ITIN as a fake SSN, but you never know. OP, I'd try and get to someone (else) in the EO who might be willing to give you more information about your particular case.
Also, I don't know the details of OPs status in the US, but if OPs officially just visiting (i.e., a tourist) I can see Cap 1 not wanting to extend US credit. (I know precisely one Canadian with a US Cap 1 card, but said Canadian was officially a student in the US--had the I-20--and employed by the university).
When my wife applied for her secured card we used my bank account for the deposit and they locked not only her card but also BOTH of my cards for fraud. After 2 weeks they had them finally unlocked and I was ready to go. Cap ones fraud department can be a major pain.
I'm curious there reason behind closing your accounts overall. Hmm..