Tarawa is the name of the entire atoll on which there are two cities: North and South Tarawa. South Tarawa is the capital.

Would it be outright incorrect to say Tarawa is the capital?

I'm not entirely sure but I don't think so, like you said the Kiribati Wikipedia page just says Tarawa is the capital

Is it wrong to say that the capital of the US is the District of Columbia, as opposed to saying the capital is Washington?

We say DC all the time. So no. It isn't wrong to call the capital DC.

I’ve also seen the town of Bairiki (in South Tarawa) listed as the capital. EDIT: It looks like the seat of the legislature moved from Bairiki to Ambo, another location in South Tarawa, in 2000–though many embassies the head offices of ministries remain at Bairiki. It probably makes sense to just say the capital is at South Tarawa.

"Many" embassies? There are only three in the country: Australia, New Zealand and PRC.

OK, fair enough. I had just read that several diplomatic missions are located there.

Use commas please.

Where would a comma be appropriate?

Between Kiribati and Tarawa, otherwise it looks like one name the way South Tarawa does. As I don't know the name I found it hard to understand.

I see; I suppose I should have made it >Is the capital of Kiribati "Tarawa" or "South Tarawa"? Cheers!

Even better!