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20" isn't that short, as compared to the standard 22" or 24". So, a change of powder/ loads may not be necessary. I second the suggestion to try your standard loads first, before deciding that you must change anything.
If your rifle had a seriously short barrel, that would be another story. My current Mauser, which I re-built and re-barreled from the standard military length (23.6" barrel), to a carbine length rifle (17.5" barrel)....would fit into the "another story" category. My standard loads with the full length barrel, (using IMR 4064 powder) produced lots of unburned powder with the new barrel, as well as a significant velocity drop (beyond that which might be indicated). A switch to a faster powder (Hodgdon Benchmark or IMR 3031) solved the issue. New loads had to be worked up, of course, but no change in bullet weights was needed. I should add that the rifling rate on the new barrel is the same as the original.
Heavier bullets "better" for shorter barrels ? I think that idea comes from the concept of trying to maintain equivalent impact energy with longer barrels (i.e., since you lose velocity with a shorter barrel....use a heavier bullet to make up the difference). It's WAY more complicated than that. Thus, this concept is simply a useless generalization.
So, depending on your current load(s), you might not need any change. Or you might need a bit faster powder, if your current powder is quite slow. I have a feeling that this will not be the issue with a small cartridge like the 6mm. On the other hand, I do not know the 6mm Rem. cartridge. But, do test your existing loads first.
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