I think the seatbelt extension is the answer. New car seems a bit OTT.
But, whilst the children are in your care, surely ensuring their safety first and foremost is your responsibility.
If I discovered my childminder/au pair/nanny was putting my baby in the front seat I would have a chat and ask them to stop immediately. I would rather they missed a routine visit to baby clinic than my baby's safety was put at risk.
I'm not sure this is the parents' responsibility, assuming you are a live out nanny and are using your car in return for reasonable mileage (42p per mile or whatever it is now). I think it depends on what was discussed/agreed in interview and contract negotiation. If they didn't ask you if your car was suitable for transporting the baby, then you probably have reason to ask them to buy the seatbelt extender. But, if they asked you about transporting children in your car and you sad all was in order, then they could reasonably expect you to show up with a car that is equipped for the purpose.
I think safety comes before comfort, and I would sooner get on 3 buses and walk for 20 minutes than put my children's health and safety at risk. The fact that this is not your priority would worry me.
But, really, we are debating things that don't really need solving here because all you need is a seatbelt extender.