Now if I were to park in a private car park, the owners of which don't clamp, but do try to get you to pay up a significant sum if you park there for more than they'd like, and if I was to change the number plates on my car to avoid this 'fine', what offence would I be committing?

Shirley private land so not perverting the course or suchlike? Not 'justice' as such if a p[rivate company tries to see you off for some cash for parking on their private land too long?

From the circs the OP describes Fraud by false representation may well apply, whether it would get past CPS is a different matter.

Fraud by false representation (Section 2)

The elements of the offence are that the Defendant:

made
a false representation
dishonestly
knowing that the representation was or might be untrue or misleading
with intent to make a gain for himself or another, to cause loss to another or to expose another to risk of loss.

The offence is entirely offender focused. It is complete as soon as the Defendant makes a false representation, provided that it is made with the necessary dishonest intent. It differs from the deception offences in that it is immaterial whether or not any one is cognisant of the representation, deceived or any property actually gained or lost.

Now we know why clamping companies got started.

True, athough if there was a legitmate looking number plate, would the private parking warden have reason look at the tax disc? AFAIK only "proper" parking wardens would have reason to check tax details...