I would try nail polish remover. I understand your desire to refrain from the use of chemicals for your toddler and dogs sake, but nail polish on it self is rather chemical.

lol this happened the other day! my child had spilt a little on the carpet and the lino—in a rental!!!!!!!!!!!!!!! try nail polish remover. i also had to go buy remover (so didn’t clean it straight away) but it works. spot test on the carpet first though. my hubby said to me not to worry because it wears off, just like it does on your nails. i was trying to rub it off and that only made the pink turn orange lol