Not sure where to put this because this isn't a job suggestion. It's just something that would affect a pet profile if the pet owner wishes to use it.
At the moment there are three options for Job information in Column 3.
My suggestion is an option for a pet's profile to read Job: Self Employed in Column 3. This is just text without a link to anything.
Pets with jobs showing Job: Self Employed on their profiles. Owners carry on as normal with getting them promotions and collecting income. (Battle pets might as well be employed because Intelligence, Strength and Level go up with training.)
Pets without jobs. They aren't officially employed so there's no income to collect.
Whatever a pet is supposed to be self employed as can go in the pet description.