According to people that want to do it humanely they suggest Castor Oil being sprayed after you have thoroughly wet the grass/mole holes down.
They also say that moles do not like the feel of human hair or the odor and they will leave, so some say that you can get hair clippings from a local barber or hairdresser. (interestingly enough, our local shop is collecting all the clipped hair to send to La as they are using it to mop up the oil spill, who knew!!
The reason the moles are there (allegedly) is because you likely have lots of grubs in your lawn, they favor those and so if you can get rid of the food source, they will leave.
Now, as far as a cat goes, they say that if you get used cat litter (kinda gross I guess) and pour it down the mole holes, you may convince the moles that a cat is near and they will high tail it out of there. Unless of course, they know its a fake out,
years ago we had a mole that was driving us nuts, he eventually burrowed under our pool liner and we have a permanent track to follow, even with all the pressure of the water, he still left a trail, now I laugh about it, years ago, not so funny! Not sure why he left, maybe he croaked under there


Good Luck! Theres lots online if you google, with little ones and pets, I'd be extra careful!