The short answer is: email the outfitter and/or the NT Police and ask the specific question.
The long answer is more complex. The Northern Territory is a territory and not a State. It does have a legislature, but the Federal Government can disallow any law they don't approve of and impose any laws they want to. Back in 1996/97, and again around 2003, the Federal Government forced the NT to adopt anti-gun legislation. Part of those laws prevented hunting being an approved use for a handgun. Because of the safari industry, and general outback lifestyle issues, the NT didn't want this but were forced to accept it. Last time I looked that's how their legislation was still written. Somewhere along the way someone came up with the idea of handgun permits "for the purpose of the safari" ie nothing to do with hunting but rather for some undefined safari purpose(which everyone knew meant hunting) Officially recreational hunting was still a no-no. The NT is one of those places where what the law says, and what the police permit may not necessarily coincide.
The next hurdle is that import approval is not controlled by the NT Government. Importation is the jurisdiction of the Australian Border Force (Customs)ie the Federal government. There may also be restrictions on calibre. However approval from local Authorities may go a long way to smoothing out the process.
There may be an quasi-official agreement in place to allow things to happen. But you wont find it in legislation etc. Its a rubbery situation and the only way is to check direct with the outfitter and the NT Police. Good luck.