Gosh OP, you poor thing. I'm sure you must be feeling devastated, still in shock, let down, angry, bitter, all those things. In that light I can completely see why you would want as little to do with your ex as possible as it will just stir your feelings up. I can also see why you might not feel you want to do anything that could be seen as being for your ex's convenience rather than just the children's.
In that light the car is really tricky, as of course it's going to make her life easier. Where it gets difficult, though, is that in making her life easier to an extent you're doing the same for your children: as Beaumur says, she'll have an easier time doing the shopping, running errands, getting them to and from school and so on; her mood will be better, and from that perspective her having a car would probably benefit your children as much as her.
The thing is, while I can see why you might think 'Let her new partner buy her a fucking car then, she's not my problem any more' her new partner isn't going to have the children's interests at heart the way you and your ex do. So while I really deeply feel for you and can see you're trying to do the right thing while dealing with a lot of grief and anger, I think under the circumstances it might be better for your children in the long run if you grit your teeth and don't fight too hard about this, however galling that might feel at the moment.
How old is your littlest? How long until s/he is in preschool? In other words, how long until your ex is able to get a job? And what's your financial situation? If you could find a way of enabling your ex to have a car, even if it's not your car then might that work?
Could you perhaps buy a car on HP for her use, in her name, and make repayments on her behalf for now in the clear understanding that you will stop making repayments at an agreed point and she will either need to give it back to the dealership or take over the payments? That way, you could have your own car back, while making life easier for your children when they are in her care, and also be in a position where it's relatively straightforward to exit the arrangement when she's in a position to pay for it herself.