Photoshop. How do you reduce colors in an image but within a certain colour palette?

Tl;dr: Is there a way to make Photoshop not only change the colors in a image into those from a custom color palette, but also only choose spesificly 15 of the colors that are in the previously mentioned custom palette?

A game I play named Animal Crossing New Leaf allows for usermade paintings. I thought it would be fun to take any of the screenshoots you can make in the game, and turn them into a painting to hang on the wall in-game. Just scale the images down to 32x32 pixels in photoshop and replicate by hand. Well, turns out it's not that easy.

To start with, the game only has a color palette of about 159 spesific colors (Imgur: The magic of the Internet ). And out of them, only 15 of those 159 colors are allowed to be used per painting. So if I were to simply image>mode>indexedColor>local and reduce the colors present on the painting to 15, I would still have the problem of that potentially most of those colors are not available in the in-game palette.

I've tried to image>mode>IndexedColor>Custom(Change the palette to a custom one I made containing only the 159 in-game colors)>then turn the image back into a RGBColor And then repeat the image>mode>indexedColor>local method. Though this has brought me varied results. Like, using the (selective) modes on the indexed color menu gave some success, but also some that turned out colors not present in the palette.

I know that I could just try to roll with it, and make the color that appeared in the photoshop version be represented by a color from the in-game palette that fits. Though I fear that doing so may lead me to a scenario where two or more colors are best represented by the exact same in-game color. Potentially worsening the quality of the image.

I guess I could also use the Forced:(Custom) option on the indexed color menu to personally choose out 15 of the colors present on the fixed palette version of the image I want, to personally make sure that the colors that Photoshop picks are the ones present. Though if there was a way to make photoshop streamline that prosess, then it would be nice to know.