I'm sure I'd answered a similar question elsewhere - can't find the post right now. Exporting the palette as a .bmp or .png file isn't practical though extracting the palette from an exported .bmp file is.
Here's one way to do it...
Using either XWE or SLumpEd - export a graphic lump (any type) as a .bmp file.
Open the file in IrfanView and select Palette/Export Palette from the Image menu.
Give the palette file an appropriate name and don't forget where you saved it.
IrfanView saves 8bit palettes as JASC-PAL plain text files that can be imported into Paintshop or the Gimp - here's one I prepared earlier.