In any event the debt stays with the airframe - which can of course eventually be seized and sold to satisfy the debt.
I've said it before, but ( like Shakespeare) it can take the repeats: Whereas mortgages have to be registered with the UK CAA, the airport charge debts do not!! Caveat emptor, chaps
If you had any inkling that the landing fees would be huge, though you had not been specifically told, then some liability should be on your shoulders. Presumably your instructor was authorised by the school to undertake landing practice at this A/D in which case - his decision, his liability; and by definition, their liability...