hmmm.....
3 issues I see:
- You are likely going to have to gear this way down to get 3mph
- The ESC you are talking about uses a servo interface. Do you have experience programming an arduino to talk to servos?
- This will be a motorized vehicle that a child will be operating. You will have to put much thought into safety. Failsafe modes. If something goes wrong, it stops operating. If the brake is applied, the motor cuts.
Are you converting an existing kiddy cart or making one from scratch?