Just a quick update:

This was happening to me multiple times on other routings. CX and BA sites would show Qatar availability but when I called they agent couldn't see it. They would refresh their system and I checked over the course of a week or two on various routings. I was not able to complete the booking online either, as it would give an error message about the credit card at the final step.

Eventually I was able to get it to book over the phone by referencing the in-process booking I had attempted. They were able to see that transaction and process it over the phone. Only then would the availability show up. But if I called fresh from the start before trying to make an online booking they did not see the availability. Not sure why it was this way, but it ended up working.