Other than the softening from whatever scaling filter used there I think I prefer the DS version, and I normally dislike super saturation approaches/warm tones.
Still have you tried pulling apart the game? At base level DS uses much the same palettes as GBA which in turn is much the same as the SNES (hence the relative ease a lot of things were carried across).
I imagine there is a bunch of stuff in https://datacrystal.romhacking.net/wiki/Chrono_Trigger
detailing where locations are, or at least eliminating enough you can readily find them. Assuming you don't just care to get a bunch of cheats and savestates and play the game to rip things manually.
The issue will tend to be that the DS will probably not use raw palettes as much as wrap them in something (see NCLR and such like http://www.romhacking.net/documents/469
). That is OK though as the formats might be supported by various tools (crystlatile2 is probably where I would start but there are others made by the sprite sheet set). At this point it is mostly matching up what goes with what, and most palettes will be named similar and in the same directory as the sprites they belong with and thus you are then left with the tedious steps of finding what is what and copying across which you do not need to be a hacker to handle if you are at this stage.
Failing that basic loop back searching is usually pretty good (palettes tending not to be compressed or fiddled with that much, if they are fiddled which you can see in desmume or something if you select update palette in real time, and if they are fiddled with for animation purposes or whatever then select a reasonably long chunk that is not changed and search for that), which is to say find the palette in an emulator, search the ROM for that same hex and it should yield the palette which you can then match to a file (in crystaltile2 it will tend to say what file this selected area corresponds to in the bottom part of the tile and hex windows).