Ideally same tyres all round is what you want, however mixed as you have suggested will work, but not ideal, especially with winter almost upon us.
As has been suggested fit a complete set of MPSS and sell your part worns.
Simples,
If you really want the car:
Visit the name on the V5 to get clarity.
Check the V5 Vin with the car.
Phone the HP company to get clear title and then get it in writing.