Do you have ANY idea how difficult it is to 'rehome dogs'..
To go via reputable rescues, you are talking MONTHS on a waiting list before there are either kennel spaces or foster spaces available. A reactive dog is not suited to kennelled rescue, and foster based rescues are much less common.
No one is queuing up to home reactive dogs, rescues are overflowing, rescues are closing because they can't afford to run. They are overwhelmed with the influx of dogs people bought in lockdown and now realise they have no time for.
You say it like you can just pop the dog in a new home tomorrow, like folk are lining up round the block to take on second hand dogs with specialist requirements. They are not. At all, they never have been, but even more so now.
So the options are to sell to a stranger, to dump, or to euthanise - for what may be a few weeks living in a car? Seriously.. never get a dog, it is a responsibility you cannot just ditch when it gets a bit tough.
Many dogs with enough exercise will be FINE living in a car, particularly through winter when the risks of overheating are far far lower. It isn't ideal, but it is something dogs can cope with, they're far less worried about where they live than who they live with in many many cases.
In teh OP's case shes moving between care clients all day, so she will be in and out of the car, they won't be alone for more than an hour, probably far far less and more opportunities for play and toilet breaks between clients than if they were left at home.