Apparently, BBPS integration is being rolled out in batches for ICICI Bank credit card accounts. For instance, both my ICICI Bank credit card accounts—Amazon Pay one and the Coral+Platinum one—have no issues fetching the requisite data from ICICI's servers on any BBPS-enabled payments platform. However, it doesn't work with my dad's ICICI Coral card. So, you should wait until your account also gets BBPS-enabled.
Also, even if you have paid the bill fully, any payments app leveraging the BBPS platform for ICICI credit card bill payments would show you your "Last total amount due" instead of current outstanding amount (Unbilled one). So, you can keep checking any time to know whether your card has been enabled for BBPS or not.
For example, your bill was generated on 1st July, and the total due amount was ₹1,500. You paid your whole dues before the due date, and also made additional purchases of ₹300 since then, then if you have been migrated to BBPS, then you will only see ₹1,500 (Last billed amount) in "Total amount due" instead of ₹300 (Current outstanding after paying the last billed amount). It won't even show whether the bill has been fully paid or not.