As a data point, I spent quite a few hours trying to book LATAM flights for SA tonight.
I gave up and rang them in the end, and was told most flights do not hold 'tax quotes'. Therefore they need to be passed on to fares. This causes the website to not show availability, or, in the worst case (?) make flights appear as available when they're not.
That said, the agent was more than happy to read me availability between airport pairs when given the codes... . Perhaps give BA a call?