Using the header information of 3 files I generated, according to Sypha, there are some inconsistencies.
The first 3 bytes are all the same, but strangely, the filesize, the number of colors and the reduced color palette number all vary slightly between images despite them all using the exact same set of colors and the same color limit:
08 00 28 18
0B 00 29 18
08 00 29 18
The blue one was missing a couple pixels on the crosses, which could be lossy compression messing with me.
Unfortunately I couldn't exactly find where the palette data lived in each image, but I have a feeling it's somehow connected to this stuff at the end.
It's a different length in each file, I'm trying to find where it starts and ends but the only clue I can seem to find is that it consistently starts around the 0000:00F0 range in a 16x16 image. It seems to end wherever it wants to, and I can't find any common patterns or anything in common between them.
The settings I used which sl0nderman suggested (they worked by the way, thank you for the tip) generated a separate palette file, that might be worth looking at.
A copy of any important files just so we are on the same page:
https://files.worlio.com/users/bonkmaykr/a.cmp
https://files.worlio.com/users/bonkmaykr/b.cmp
https://files.worlio.com/users/bonkmaykr/c.cmp
https://files.worlio.com/users/bonkmaykr/a.p11
https://files.worlio.com/users/bonkmaykr/b.p11
https://files.worlio.com/users/bonkmaykr/c.p11