Yeah, didn't know why I said Lindblum. But that's still unjustified, due to the fact that Cleyra wasn't doing anything but minding their business in their desert city. They weren't enemies of Alexandria because they weren't doing anything toward Alexandria, which is why testing Odin on them was unjustified.
You forget that many Burmecians retreated to Cleyra, including the King. Thus Brahne's argument would have been that Cleyra was harbouring those who would plot to destroy Alexandria, and so her only option would be to destroy it utterly.

See this quote from the game:

Quote Originally Posted by Final Fantasy IX
Brahne: "This is a great victory for us. Kuja, your black mages made
this conquest so simple. My only concern now is finding the
king of Burmecia. We must take care of him once and for all,
and prevent these rats from ever rising up again. Beatrix,
what's taking so long!?"

Beatrix: "I don't know, Your Majesty. I've ordered Zorn and Thorn
to search the preimeter, but there's been no word so far. I
will join them and lead the search right away."

Kuja: "You're wasting your time."

Beatrix: "What?"

Kuja: "Rats often look for new homes when they sense an earthquake.
They probably moved to the sandy tree house by now... So, you
see, it's too late. The king has already turned tail and fled."

Brahne: "Sandy tree house... Surely, you can't mean Cleyra!? It'll be
quite difficult if they escaped to Cleyra."


So assuming that Beatrix believed Brahne's claims that the Burmecians were plotting against Alexandrians, it would have been difficult to argue against not attacking Cleyra at all. The use of force was obviously of concern to Beatrix but at the end of the day she realised that she wasn't commander in chief. Turning her back on Brahne at that moment would've been very difficult.

It was only when undeniable proof that Brahne had gone corrupt was offered that Beatrix changed her mind. This proof was the maltreatment of Garnet, someone who she knew to be innocent.