For BBPS enabled cards, apart from HDFC, I'd recommend Slice. HDFC pays the bills appropriately 4 days prior to due date. Slice pays just a day prior to due date.
If your CC issuing bank supports NACH, then NACH gets triggered only if it is not paid on the scheduled due date that may be different from actual due date. The amount that you prepay will be deducted from the due. If the due is nil, auto-pay won't be triggered.
Example: OneCard supports NACH auto-pay via multiple dates (before the actual due date). Axis supports NACH auto-pay, but never used.
ICICI doesn't support NACH.
For others, please check with the card issuing banks for the support of NACH auto-pay.
I am not sure how auto-pay in BBPS works (whether it deducts the prepaid amount or not). Needs to test it out in the coming months. If anyone already tested it, please share the details. Thanks.