In legal slang, you're trying to "pierce the veil" -- you're looking for a way to get around a legal impediment. You really need a lawyer, who knows the laws of your state, and the way the courts have interpreted those laws in trial cases, to get the best answer.
Generally, unless the current employer has deep pockets, they won't challenge you, if you're not aking their clients away from them.
Read your non compete clause carefully -- if it says you won't "start a business", you might hire on as an "employee" of the clients, so that instead of you being a business which sells them dog walking service, you are instead an employee whose duties include walking of the dog, in addition to other duties like shopping, washing the car, etc. To be an employee, you must work specified hours for them, they must withhold income tax and social security taxes, must match social security, etc.
How long does the non compete clause run? Is it for a period of one year after leaving their employ? Is it only in effect if you quit voluntarily (if they fire you, does it still apply)? Perhaps you might do another type of work for a time, then start the dog walking business part time (after the non compete expires), and grow it into full time gradually.
The fact that you don't like the way they treat their employees and their clients isn't strong enough reason to allow you to ignore the clause.