This not remotely a "PR nightmare" Passenger ticket problems occur routinely and most people who read this just roll their eyes and blame some Iranian bureaucrat, if anyone.
As this is all third or fourth hand reporting and the entire conversation is not reported, the full facts will likely never be known. But, if the reporting regarding the infant's name mismatch is accurate, it is possible that either AC or BA loused up the name on the infant ticket such that it did not match the infant's passport. While that might have been missed on the outbound segments (or was correct in the AC PNR, but not BA's), the BA agent in Iran caught it and refused to board the infant. One would hope that the parents did not leave the infant behind, did not board, had their tickets forfeited and then purchased new tickets back to Canada.
If this was the case, it had nothing to do with the infant as an infant, simply as a passenger. No way to tell here who, if anybody, loused this up.