Yes and no I'm afraid.
The backgrounds are in 16-bit colour. However they don't use the *full* range of 16-bit colour, it's paletted. Like I said, a few thousand (of the ~65000 colours 16-bit gives you) are actually used.
Most paint programs use 256-colour palettes. Why? One byte can store a number from 1-256 so each pixel can be only 1 byte (8 bits) but 'reference' a 16 bit - or more often 24-bit - colour.
*NO* program supports a palette with more than 256 colours. Why? The palette index each pixel held would take up more than 1 byte. And if you're going to take up 2+ bytes for each pixel then you might as well just store the colour value itself!
What Square probably did was render the background in full 16-bit, un-paletted images, then when the images were converted into field files, work out which colours were used and build a palette.
What THAT means is that the palette can't be exported as a standard 256-colour palette file. Originally the images *weren't* paletted. Now they are.
You may be thinking, why can't I convert them back into 16-bit images, then when you import them, palettize them again. The answer to that is I haven't written the code yet! Yes, it's possible.
OK ... do you understand now why I can't export the palette as a standard 256-colour PAL file?