Oxford do not care about exam mark schemes or A level criteria so if you feel the competition essay illustrates your skills most strongly, send that in.
Equally, assessors will be aware that different pieces of work are trying to do different things and will mark them on their own merits. Just as they would not expect a timed piece to be the same as a vacation essay. If the piece of coursework is a great example of its type, they will read it in light of whether it is successful in what it sets out to achieve.
Don’t write something new, though, they want to see original work, not something made up for Oxford admissions.