Have a read of this. Hopefully, what you need is in there, in addition to the help already offered. Not hugely sure that a private transaction document will carry much weight as any document provided could have been created by you. A commercial invoice can be checked and has a number of legally required elements to it, which are traceable.
That's not to say that you shouldn't take up the offer but never volunteer anything that is not being requested (a general rule of thumb). Not trying to be difficult but just raise an awareness.
https://www.gov.uk/guidance/transfer-of-residence-to-great-britain