The flip answer that came to mind was "bad hands".
No, seriously, Gums or lips? I suppose salt water is the only thing I can think of, but more to the point, why does the mouth need to be hardened?
Maybe a change of bit might work best......softer mouthpiece, eggbutt rather than loose ring etc, slightly wider perhaps? No bit at all?
I wouldn't want to be doing that! Why is it getting sore at the edges with the bit because its a bit dry?
In that instance I'd put grease on the lips when I tack up.
I use hemaroid gel on the sides of my coloureds mouth to shrink the skin after he has had a cut- it heals faster this way. My friend used to put white spirit around the muzzle to stop bit guards rubbing.