Sounds like puppy's stomach
doesn't get along with the toothpaste. The guess here is that the toothpaste is creating a stomach irritation, causing increased stomach acid, and the regugitation of stomach fluids. It's doubtful the chews are causing the problem.
Try stopping the toothpaste for a few days to see if things clear up. Once they do, give puppy some food first before trying to brush the teeth. A full stomach is less sensitive to acid irritations induced by other things.
If I only had such a dog...
My two monsters eat their dinner (or breakfast) in about 20-30 seconds flat! They barely taste it.
The key I believe will be to get something in the puppy tummy before attempting to brush. Dog treats, whatever works. Try it and see what happens.