Hi Martin,
This subject is a case of 'whatever works for you'.
It's all a matter of your little person's ability to grasp the 'push one to go, push another to stop' theory.
Looking at the website you have linked to, you could use the 4" band type brakes, and operate these with a cable, and devise a system that could operate a rod for the Hand / Parking brake on the same band without too much effort.
As the car will only be moving at quite slow speeds, this should be more than adequate for the job in hand.
Personally speaking, I have used the 'Electric Brake' which is standard fitment on the more modern Mobility Scooter for the time being, as I felt that this was easier or my then 3 year old Grandson (he's 4 now) to use (push to go, lift to stop), as this not only gives very adequate stopping power, but also effectively applies an 'Automatic' parking brake system.
This said, I have also made provision for a 'Manual' braking system, and mounted 'Mountain Bike' disk rotors to the inside of the rear hubs, and bought 'Mini Moto' calipers to provide the stopping power. These will be cable operated, applied from a 'Pendulum' pedal mounted on the bulkhead.
Just at the moment, the 'electronic' brake is fine, but I can change to the 'manual' system when Harvey gets a little older and more co-ordinated, without too much trouble.
As Richard say's, use your own judgement, and maybe you will find a simpler solution for others to use.
Keep Thinking,
Regards,
Mark.