Hi jlb welcome to the forum. I would probably try the camping route I would imagine it would be the most economical. Certainly some of the smaller campsites that we have viisted in the past have accepted campers any day of the week.
The first camping trip we did was to
Cloud Farm in the Doone Valley on Exmoor. They are very flexible and although they do not accept bookings the guy who runs the site assured us that they had never turned anyone away. It is situated right on Exmoor and next to a river which is great for children of all ages and adults too!
We are usually very unorganised and rarely book anywhere in advance but seem to always muddle through sometimes if you call particular campsites that happen to be full the owners can often recommend somewhere close by that may have vacancies.
As far as the car situation you could always get a roofbox or failing that most children are happy to be a bit squashed if they are going holiday! We have travelled in our very small car with our own two girls 4 and 6 our 13 year old nephew and a large greyhound (I think she was probably the most comfortable)...