Timatic was created precisely in order to provide an authoritative record of what the rules are and I believe airlines pay good money for access to the system. Internet websites, on the other hand, are not authoritative, especially when we are talking about infrequently updated, English-language sites put up by countries that don't exactly rank high in the transparency department.

Unless the CoC says certain internet websites will be considered final and binding regarding the entry requirements for certain countries, I say fight it out with the airline as CoC does not allow the airline to deny boarding without compensation based on a subjective opinion of what check-in agents think the rules are.