I'm not sure about training it that way but I feel like this is where "drop it/leave it" comes into play. Puppies put everything in their mouths i've come to learn...rocks, shoes, sticks, bugs, cats, hands, hair. When it comes to food though we don't ever feed Pike from the table or anything we are eating ourselves so even if something falls to the floor he doesn't seem to seriously care about it - and I think that's because we don't allow him "people food" so he has no interest.
I think leave it/drop it would be more successful for you- that way even if something is in his mouth that you didn't get to in time he will know to spit it out immediately. I accidentally used "let go" in the same manner when I was training in place of drop it and it works the same, he'll spit out whatever is in his mouth, even a toy that he knows is okay to chew. It's a handy phrase to train.
Hopefully that helped some. (: