They should arrange for the car to be collected, especially if you are not confident or happy with the possibility the car may cut-out whilst in traffic or leave you stranded...or if you are a lone female driver.
The charge the mechanic is placing in the battery will only last for a limited time, if the alternator is not charging. The car cutting out is inevitable, it's just a matter of when. If you explain this to the dealer; as a good customer relations exercise they should arrange the collection of the motor.