My thoughts were to make it a simple 16 colors (not 16 bits) images
In general, there are no such things as 16-color images.
The closest you'll get is palettised images which use 8 bits per index (3-bits each for red & green and 2-bits for blue) giving a 256 colors in total.
Of course, you do not need to use all 256 palette entries. If you only use 16 colors, then the size of the palette stored in the image file will (sometimes) be reduced(*), but the image will still require 1-byte per pixel to store the indexes, so the gain of going below 256 colors is minimal(**).
(*From approximated 4*256=1024 bytes for 256 entries; to 4*16=64 bytes for 16 entries. Not all image tools do this.)
(**There is the possibility that using a palettised .png that only uses 16 colors might compress more easily thereby further reducing the size of the image file.)
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