Assuming no derogatory info based on scores? The Amex Platinum card should be about as close to a guarantee as you could get in the credit world; a denial for it would be surprising. How old are your other cards? The CSP will require a minimum credit line of $5,000 and your other two cards are right around that. Chase usually (but not always) likes to see a bit of history handling larger credit lines. If your credit profile is quite young, I might consider holding off on CSP for a bit but keeping in mind their 5/24 policy.
Since Capital One and Discover both offer credit line increases without a hard inquiry, have you asked for increased credit lines on those accounts? Just to make sure, by "zero balance owed," do you mean you are paying the statement balance in full each month for both cards or are they both reporting $0 to the credit bureaus at statement close? Unless at least one card is reporting a balance, you will actually lose points in your credit scores since it's riskier to not use revolving credit at all than it is to manage it responsibly. Doesn't mean you need to carry a balance and pay interest, but at least one card should always report at least $5 or so each month.
