Welp, I gave it a fair try. I've been using Biotene toothpaste and mouthwash for just under a month now. Since starting Biotene, I've had two canker sores on my lower-left lip. The first appeared after I ate hot, acidic food (pizza), so I gave Biotene a pass. The first has since gone away. But, when a second sore appeared, I wasn't willing to wait any longer. When using Trader Joe's peppermint toothpaste over the past 3-4 years (and eating a wide variety of foods), I've seen canker sores very rarely and never two in such close time proximity. I suspect that the problem is the detergent in the Biotene toothpaste (Isoceteth-20), but also wonder about the mouthwash, which contains propylene glycol. Propylene glycol is a possible cause of contact dermatitis.
Update 7/14: My remaining canker sore is almost completely healed 2 days after I switched from Biotene back to Trader Joe's toothpaste.