There sure are some dodgy characters out there. I have often helped classic car owners by moving their cars with my big flat-bed trailer. All I ask is that they give me a few quid to pay for the diesel used.
With a Mini, or any classic car needing work, the only real thing to do is to learn to do-it-yourself. Classics are time-consuming and at between £40 and £60 per hour they cost a fortune, even if you can find an honest expert. As an example, my Grandson bought a 1997 MPI a few years ago for £1600. It was 'cheap', but we had anticipated that the 'few small rust bubbles' would lead to a lot of work. In the end it needed a new screen scuttle, two front wings, a door skin, both A-panels, a front panel, one outer sill (with a small repair to the jacking point), inner-wing patch repairs and some odds-and-ends. The engine was removed, new bearing shells, new oil pump, new cam followers, head cleaned and valves re-lapped, centre oil pick-up fitted, new lay-gear bearings. The brakes were overhauled and new suspension rubbers fitted. Total cost for parts was around £850 plus the cost of the paint for re-spraying the body (not the roof). If we had paid for that work it would have cost over £3500 at normal commercial rates.
DIY is the only way.