I chew mint gum sometimes while fasting. I have done a lot of MC's, mostly by the letter, but also a few modified versions. Chewing Gum or not chewing gum, not a problem for me either way. Almost everyone gets what I call "death breath" when fasting. Get some no calorie gum if you do decide to chew.
And remember, if you chew gum, you might make yourself very hungry due to the stomach getting signals that food is on the way. It doesn't hit me that way, but beware.