Lawyers could probably convene lengthy conferences, with contribution from dozens of members of the legal profession, to argue over the differences between 'public' and 'private', without any final agreement ever being reached! ;-)
A couple of examples might be relevant here:
1. If you were to have a knife inside that caravan, while it was parked on the highway or on a car park (irrespective of whether that be a 'public' car park or, say, a supermarket car park) you would risk being charged with being in possession of a bladed article in a public place. (The fact that the caravan is, in itself, 'private' is completely irrelevant. The location of the caravan is what matters).
2. Homosexual acts between consenting adults are, of course, legal as long as they take place in private. You might think that a prison cell would be regarded as 'private' (since the general public clearly have no access) but the courts have ruled that it is illegal for prisoners to engage in homosexual acts because prison officers may observe such activity.
In practice your problems might not be directly related to the specific nature of your business but to the general laws (or bye-laws) which apply to all businesses. It's usually an offence to carry on any form of business on any car park (or similar area), owned by a local authority, without the consent of that authority. Similar rules (on 'street trading') normally apply to operating any business from a vehicle parked on the public highway.
Chris