Mint can grow pretty tall (about 12") when it's flowering. It not only will survive the dogs, it'll creep into the rest of your property and be impossible to get rid of. It's extremely invasive. Your dogs will always smell of mint. Consider all this.

And you'll live to regret that landscape cloth. It's a nightmare.

Ajuga may work. It also can be invasive but at least by reputation, is easier to control than mint. It is also recommended as an erosion control plant, and is commonly available and easy to reproduce.