If it were me, this is what I would do:

• Paste the recolored image back into Illustrator
• Select an individual color in Illustrator, then go to Select > Same > Fill Color to select all the pieces that have the same color, then
• Use the eyedropper tool to apply the Photoshopped colors to the original image.
• If you’d need them to be PMS colors, (it may not be exact) I would use Edit > Edit Colors > Recolor Artwork, and apply the specific color book you want the colors to pull from.

Hope that helps!