Update - There are two pinion gear kits for the GX. One for 2010-2014, and one for 2015-forward.
2010-2014 kit is 41201-80195
2015-forward is 41201-80785
No clue what the difference between these is, I assumed it would be the same part from 2010-forward. Alas.
To play it safe I restricted my search to the 2010-2014 part to fit my 2013. This part is difficult to find - called a couple of dealers, one dealer says that the part is backordered in all of North America. Another dealer says there might be one available in North America, in Portland, OR. Part runs about $1,000 + labor. The risk with going this route is that it does replace the main gears in the diff, but not all of them, so if any other gears outside this parts kit are contributing to the noise, we won't know until the diff is back on the car. So this route can get expensive depending on which gears are at issue.
Alternatively, a salvage diff can be sourced, or a new one can be installed:
Salvage diff of course comes with its own risks of other issues and no warranty. Salvage diffs of this exact part number are also relatively scarce as 2010-2013 GXs didn't sell as well as 2014+ GXs. 2014+ (which I think will make the 2010s-2013s collectible in the future, as the purest looking "Prados" available in the US). Salvage diff route at this point in time was quoted a little over $2,000.
New diffs come with no wear and a two year warranty. New diff route is the surefire way to resolve the diff issue and comes in at about $1,100 more than the salvage diff route.
Not a cheap proposition no matter which way you slice it, each will have to make his call based on his means and use case. For me, this sound is the difference between me keeping the vehicle forever or dumping it in the near future, and I want the sound gone yesterday. I do plan to keep this truck forever, so I am inclined to just replace it with a new diff and be done with the issue.