1. Iroh was still encouraging Zuko to fight his sister, because they couldn't risk Azula being in a place to threaten any new government or continue the old.
2. Katara going along still doesn't really solve the main issue at hand - that the people would still be seeing it as a coup by Iroh and Zuko. All Katara's presence would mean it was Avatar backed, which will help those outside the Fire Nation accept it, but not those within. That said, it is more important that the outside realize that than the citizens, to smooth over the peacemaking process. The FN citizens would be expected to fall in line.
3. Zuko didn't head to the Fire Nation with the intention to fight an Agni Kai, and whilst you could argue that's how Iroh expected things to play out, his claim is no more legitimate than it would have been beforehand.
There's nothing that could have been done that would prevent the people from seeing the country as a coup. All they could do was run damage control, limiting the Fire Nation's ability to continue a war post-Comet and hinder Ozai's plan to raise the Earth Kingdom.