Update: I called UK line and got it ticketed at exactly the same fares advertised online (before error message). The agent did not have to go to Helpdesk for pricing, but still took good 30 minutes since I do not have BA profile so everything had to be repeated over the phone . They waved phone agent fee, because website was not cooperating.
I am not buying “complexity” issue as excuse for manual ticketing, it should be the opposite. I work in IT and also understand how complex fare structure is. I have seen many agents make mistakes and halve hard time interpreting complex fare rules. This is exactly why software should check all rules much more efficiently, unfortunately the software was written in 1980s and airlines did not invest enough in software upgrades.
With respect, I think you are underestimating the complexity of airfares in genera, and the one you specifically bought - which appears to be borne out by the fact you needed half an hour with the agent to book it.