I’m little surprised to hear about your issue with Citibank and BPay. My experience is completely the opposite. I do recall some cards’ T&Cs excluding BPay but I’ve only ever failed to receive points when using for BPay when it was to the ATO with Citi. I agree with @darrenk comments about card fees, cash advances, etc though I haven’t actually tested those. What I can confirm is as follows (all QF points -please forgive the prevalence of QF):
Citi Signature Visa (QF): Account opened >10 years ago. Converted to Signature ~ Nov 2011. Last received points for BPay Aug 2015 for school fees & roadside assist.
Citi Signature Visa (QF): Opened Sept 2015. Last points received Oct 2015 for local council rates, VicRoads rego & Online betting (don’t ask!)
Jetstar Mastercard (QF): Opened June 2015. Last points received Aug 2015 for BPay to ATO.
ANZ Black (QF) : Opened April 2015. Last points received Aug 2015 school fees
HSBC QF Visa (QF): Opened Oct 2013. Last points received Sept 2015 for BPay to two separate online betting companies. This card has now been converted to HSBC Rewards so we’ll have to wait and see if it still earns points for BPay.
So, overall I would expect to receive points for all “normal” BPay payments except to the ATO. Perhaps try a different card? If you have ATO payments then the current Jetstar offer (or HSBC for flexiblity) otherwise take your pick!
(Name withheld for embarrassing online behaviour including collecting QF and online betting)