So I presume you've set out all the options and pros and cons of each option?
Try to stay where you are, rent out a room to a lodger, dont assume your bills will go up to £700 a month, they might not, see if you can move to interest only
Is your A level son over 18? Can he move out on his own accord and claim benefits, there are circumstances this applies, the disadvantage would be that you would no longer get CB or other benefits associated with looking after him
Do your teens have part time jobs, can this provide contributions to their food bills (not ideal, reeks of Dickens but keeps them in their own home), would this even be enough to survive
If you move to this free rental place, how does WFH work if the wifi is ropey, what about heating bills up there, are there problems with having 'free' accommodation in terms of your family members tax implications as they are still a landlord. What about petrol costs to get anywhere, huge journey times. No schools for the kids, no friends, no hobbies. How long would you stay there?
Option of moving to rented near where you are now, using the equity for your rental payments, what are the pros and cons to this, is it feasible, agencies will not often rent to you if you're going to be overcrowded so its hard getting a small place and deciding the parents sleep on the sofa, you wont be allowed to rent that property. Do you have a guarantor if you have a poor credit rating
Option of getting a mortgage with a provider that is designed for higher risk borrowers, the disadvantage is the high rate but I paid 8% on my payments back in the 90s, it seemed normal then. Then you get to choose the size of the home, doesnt matter if its too small, also build up your credit rating again and change to a new mortgage when its repaired, bills will be smaller, kids get to stay where they are