This is fairly simple:

1. call to_paletted() with the quantization options you want, in your case max_colors => 16
2. call write()

For example

my $pal = $source->to_paletted
  (
   max_colors => 16,
   translate => "errdiff",
  ) or die $source->errstr;
$pal->write(file => "foo.png")
  or die $pal->errstr;
[download]

That said, unless your images start with a small number of colours (non-photographic, no blends, little anti-aliasing) the results are going to be fairly ugly. You should test with some representative source images.

pngcrush afterwards?

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.)

It seems that Imager supports dithering. At least Imager::ImageTypes mentions "Floyd Steinberg error diffusion", which seems to suggest that it can be used to further reduce the number of used colours at the cost of image resolution.

I think you'll find that once you write the images to disk in any of the popular (gif/jpg/png) formats, they'll be stored as 8-bit indexed palletised images.

The tiff file format might theoretically support 4-bit indexes; but I've never come across one; nor any tool that would write one?

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The start of some sanity?

Looking into it a little further, it seems that Imager can write Windows .BMP files with 4-bits per pel.

However, as it doesn't support compressing those images, the file sizes will likely be much larger than the same image written as 8-bit palettised in .png, .gif or .jpg format

Imager doesn't compress BMP files at all, though it can read BI_RLE4 and BI_RLE8 compressed images.

It never seemed worth the effort of implementation, with so many formats with useful compression algorithms.