This is how the vanilla TSA arranges the palettes if you want to work around it:
(blue is palette index 0; red is 1; green is 2; purple is 3)
Here's how a custom statscreen that utilises it looks in Usenti:
(only the first four rows of colours in the palette are necessary— the rest are garbage)
and here's how it looks in the game:
For what it's worth, I would suggest just using a 16-colour image with the same palette on all four rows unless it absolutely has to have a bunch of colours; it took a lot of tedious trial and error for me to get mine to behave.