Depending on whether the perlino is homozygous for black and bay or not, and whether the chestnut has any bay genes hidden:
Also, it might be possible that the Perlino is hiding something like dun, which could also be a factor.
Genetically, there is no known difference between a liver chestnut/sorrel and a red chestnut/sorrel and a light chestnut/sorrel. It's just a different shade of the same colour. If the foal comes out palomino with a liver chestnut base, it may appear slightly different than a palomino with a red chestnut base.