For future historians (and chasers of US CCs), if you do not have an SSN or have only recently obtained an ITIN, building your portfolio up with AMEX US (personal) cards is highly recommended. The safest option is to apply the 2nd card with your ITIN, wait at least a year after receiving your FICO score (6 months + 12 months) through AMEX, and then start looking at the market for what is available.
If you do have some sort of bank account in the US, it will also help somewhat collate your info (esp if you applied and accepted with your ITIN)! I do not know the requirements of the AMEX Savings account, so there is no DP there. Additionally, around the 12-month mark, you could try looking for offers that are like 'check to see if you're pre-approved' as it will also give some indication of what banks might be willing to take you. But as of writing this post, do not apply for CREDIT ONE cards, which can commonly be mistaken as Capital One's credit cards.