Oh god no - it sends a shiver down my spine to see a dog carrying a stick.
Many years ago, my first dog got a stick caught in the back of her throat whilst on a walk. She could barely breathe and it was the most horrific few minutes and myself and my partner tried to free the stick from her throat. As we were in the middle of nowhere, getting her to a vet was not an option. Eventually it was sheer luck that I managed to get my fingers down far enough to hook the stick and manipulate it round so that I could free it from her throat.
I would try and create a "special" toy with my dog. One that's owned by me and she is allowed to "borrow". I've created these for pups in the past by running round the house having lots of fun, showing her the toy but not allowing her to play with it. After some time, I "allow" her to play with the toy for brief moments before putting it back on my shelf and reclaiming what is mine

On a walk, I would slip this in my pocket to produce when I needed immediate attention.