YANBU- this sounds downright sneaky.
BUT 5k is a drop in the ocean and your financial situation sounds absolutely terrible - you’ve both got to stop living hand to mouth. Driving a potentially dangerous car is a big deal. You’re both healthy, you have one child in school, this is your time to save money and prepare yourselves for the future. You can’t not work for a huge portion of the year and then be surprised you’re poor. When I was in my 20s/ early 30s I was pulling 60/ 70 hour weeks!
I know others have given ideas but you need to get a job which pays more and which pays during the holidays. Schools are crying out for teachers near me, especially those with SEN qualifications. If you think that’s not possible then start thinking of other options, you must have a degree and transferable skills. What about being a PA, marketing, office admin, getting a HR qualification, working for the local authority, becoming a school governance professional? What have you investigated?
In the meantime and especially over the holidays I personally think they each of you should take a non-demanding second job. Like a couple of shifts on the tills at a supermarket over the weekend, driving a taxi (he could use the electric car), tutoring with an agency, exam invigilating. Lots of very middle class people I know have done this when money was tight. If you each bring in an extra £50 a week, that’s £400 quid a month.
You should also be prepared to make some drastic changes- like moving area, changing where you rent (rent a 1 bed), moving in with family, so that you can start building up some proper savings.