Is this car still registered? You should be able to find out using your traffic authorities’ web site.
Regardless, if you tow the car outside, and leave it parked normally in the street (not on the nature strip), this should not be immediately considered “illegal dumping”, and I doubt the council can take action against you. It would just be an unregistered car parked there. Even councils have to wait a reasonable period of time before the car can then be declared “abandoned” and remove it, which becomes their responsibility. It seems that they cannot claim compensation from previous owners even if they are identified. The original owner is hardly likely to come back and complain that you moved his car.
In the past I have had a few cheap old “bangers” which I have used till they break down. I have then rung up one of the companies that offer “free removal” of cars; a tow truck turns up and takes the car away, I have never been asked for PoO or even personal ID.