According to the picture in the review, I confirm it's an MLX90333 3D Hall Effect sensor chip as I have guessed. Here's a picture of my circuit board for DiXYHS, designed to work with Cougar (I get to keep the old badly soldered one and my partner in Australia gets the more refined one). From the picture, you can see that the first row of the marking is 333BCM, the same as in Joe's review picture (thanks Joe for the close up shot!)

So, the SPI (Serial Protocol Interface) output is indeed 16bit, but the resolution is only 14bit, according to MLX90333 Triaxis 3D-Joystick Position Sensor specification, pp. 16,
http://www.melexis.com/Asset/MLX90333_Datasheet_DownloadLink_5276.aspx. Meaning, you get 16bit of data per reading, but there are only 14 bit resolution scaled up to 16bit.
So, please don't follow TM's marketing line and say it's 16bit resolution (it's 16bit output, 14bit resolution; the marketing people might not be able to tell the difference, we techies could and should), unless you tell me TM's got a more advance revision of the chip (not likely, because the "BCM" in 333BCM is the die number, pp. 42 of the same spec. sheet). MLX90333 internally has a 15 bit ADC for reading the raw data (voltage induced by magnetic field), so it is impossible for it to generate more than 15bit of resolution. In fact, 14 is what you can reliably get out of 15bit according to Nyquist Sampling Principle.
When I get my paws on the piggy, I will report back what the internal chip programming is and whether the chip is locked from reprgramming (no big deal, I can make new ones). Why would you want to reprogram it? Oh, well, if you want to make a different mechanism like a center stick configuration, you might want to have different response angle range, thus different response curve, say I want all the 14bit resolution to be between the angle of +-20 degree instead of +-35 degree. You can dial that in the MLX90333.
Also, the PCB looks like it's sitting in a recess slot, but I can say in relative confidence that you can jack the PCB up a bit and rotate the sensor, as long as the censor's center still sits at the same location, you should be fine. MLX90333 is not very sensitive to the magnet distance (actually the strength of the magnetic field), mine works fine between 0.25" to 1". Looks like you center stick folx are in luck.
Hmm... double sided PCB board for the MLX90333 in the Warthog.... it took me some time to route everything on one side so I can just CNC mill the PCB in one shot w/o having to flip side, align, mill again, and plate via holes.
Now, I am sure I can mod it to my liking.