First off, before we dig into this, let's get the criteria out of the way: Whoever signs up has to first meet the approval of author Suzanne Collins, who wants a solid filmmaker to take the job. They also have to have their schedule clear and be able to start work in August, the chosen director also must work on the script by Collins and Simon Beaufoy, as well as do additional casting, pre-production, and a million other details. And it will require someone who won't ask any questions, just put their head down and get to work.
So let's start with Cuaron. It's hardly a surprise he's on here, as fans have been trumpeting his name thanks to his work on "Harry Potter & The Prisoner Of Azkaban." But here's the thing: He's in post-production on his big, ambitious, sci-fi movie "Gravity" which has a zillion special effects, and it's likely something he's going to want to be tinkering with and finessing right down to its November release date. Not to mention that he will be heavily involved in what will probably be an extensive press run for the movie, which will likely clash with the "Catching Fire" schedule. We don't see this happening.
As for Cronenberg, as inspired as that would be, unless he's aching for a paycheck gig, we don't see him being at all interested without an opportunity to put his own personal stamp on it. That said, the concept might be odd enough to intrigue him, but enough to tie him up for the next year-and-a-half and keep him away from his own projects he has brewing? Again, we don't see it, unless he strikes a one-for-them, one-for-me deal with the studio (that said, Lionsgate probably don't want to sink any more money into a director than they have to).
Oddly enough, Inarritu is entirely feasible. Why? Firstly, his "Babel" producer Jon Kilik is working on this franchise, and secondly, his calendar is free. While the period drama "The Revenant" and the true crime pic "Flim-Flam Man" have been percolating, they haven't been moving forward. Taking on a franchise movie, knocking it out of the park and putting himself back in the spotlight could get those things moving again. And at the bare minimum, it would be a lucrative job. Plus he has the cinephile credentials that will make both the author and the rabid fanbase happy.
The Lionsgate list is apparently topping out around eight names, and we have our own suggestions too, but at least these names indicate they are trying to think outside the box on this one.