I did read all your posts before replying. You said a reactive dog - that seemed to be in response to people saying about staying with relatives. I didn't see anything that mentioned that meant she needed a garden. I didn't realise that was what you were referring to.
If she's not going to cope in a house without a garden, she's going to struggle even more in a car. Especially if well-meaning passers-by bang on the window when you have to leave them in the car unattended. And to be fair, depending on where you can find to park up, they might bang on the window when you're in the car too. What about when you go into the gym for a shower/toilet etc? I don't know any gym that allows dogs in. And hospital appointments etc. It will be awful for you and them.
Btw, I don't blame you for not wanting to give up your dogs. I wouldn't either.
I read above that you're autistic. I'm also autistic. I completely understand not being able to share your space.
Please don't think I'm unsympathetic. It's an awful, awful situation for you.
I cross posted with your comment re autism, but I think sometimes we can be too rigid without realising it. I was replying to your comments on this thread turning down houses because there was no garden/too far etc. I can see the main issue is with being accepted, and that's hard to overcome. I just meant that I would apply to houses without gardens, further away etc even if they're not ideal because you/your dogs are going to find it unbelievably hard in the car. Of course that doesn't mean you'll be accepted, but it might give you a larger pool to pick from. Don't hold out for the house you really want.
I take your point about saving money but the worry is about getting to that point - for both you and your dogs. I'm genuinely worried about how you'd all make it that far.
Housing in this country is in a terrible mess right now.