How strict are the color requirements? Some engines (looking at pico-8) can accommodate the resolution but not the colors EXACTLY (see example image). Given that people are already playing it a little loose with things like pixel fade and such, can we get away with being slightly off-color?
I don't know the actual answer here, but you could make the game as normal in PICO-8 and change the colors manually for the HTML export that you submit to the jam here on itch (it's easy enough to change the color vales in the exported JS). And you can approximate the colors better than that using the extended secret palette (one of the secret colors is #125359, for example), although you obviously can't get them exact.
I just checked, it is indeed easy enough to just hack the JS output and swap the colors for the desired jam colors.
For pico-8 devs who are interested - after exporting to html, open the generated javascript file. The pico-8 color palette is stored as RGB values, comma separated, no spaces. If you want to replace RED (index 8) with another color, just search for "255,0,77" and then swap these with your new desired RGB.
Thanks!