It sounds like there was a miscommunication between OP and the phone agent.
OP heard from the agent that it was okay to change the billing details for the very same ticket (ticket number and PNR) OP booked.
Agent thought OP was asking if it was okay to cancel the ticket within 24hours and pay for the same trip (but not same ticket number and PNR) using a different form of payment.
It is a tough call, but unless OP can prove that the agent gave wrong information, I think OP is out of luck. I can understand how the miscommunication occured, but, unfortunately, this is just one of those lessons we learn as we become frequent flyers.
Bottom line: once you book and pay for a ticket, and the ticket is issued, even a refundable ticket, you can't change the form of payment for that ticket number/ PNR. You can get a refund and re-book, if the ticket is refundable, but you are not guaranteed the original fare.
At the end of the day, OP got a good deal on the ticket, and will be able to use the miles/points on a future ticket.