Does having a CC with them consider as banking relationship? And not to be mean/racist but I can't really understand those CSR...
No, need to have a checking/savings account etc. But as everyone has said, pushing from your bank's bill pay is the way to go, to get round any limits on the issuer's web site (e.g. FIA limits the number of payments you can make online a month, pushes from banks don't count towards the limit)
In general, I've never had any issues with making payments with Citi online. Not sure what the issue might be with your account OP. IME, the minute the transactions post to any of my Citi cards, whether it's the Divident WEMC, AA Amex or TYP, I can simply make the payment right then and there and is not dependent on the previous statement balance, since I PIF and it's always 0.
If this is your FIRST statement to generate on this account Citi will not permit an online payment UNTIL a statement is generated. You would have to go into a branch and make the payment there by check or cash. I have 3 Citi cards and each one of them this has been the case.