Thanks to everyone for the feedback. A few comments from my side:
(i) I believe that the 300 ZX Convertible has a kerb weight of 1500 Kg ? To put that in perspective, a Porsche Carrera 997 Carrera S Cabriolet is also 1500 Kg. So is a Ferrari F430 Spider. The Lamborgini Mucielago is 1690 Kg. Even the latest Porsche Boxster is 1345 Kg - and that is light by modern car standards.
Does this mean that these cars aren't 'sports cars' ? I have heard that, if you are a purist, a true sports car should not weigh more than 1000 Kg kerb weight. That means even the TVR Chimaera and Griffith (around 1059 Kg miss out. In fact, the only two that sneak in are the 1999-2004 MX 5 (1000 Kg, I think) and the MR 2 (a tad under 1000 Kg).
My interpretation is that the 300 ZX is closer spiritually to a 997 etc. (i.e. a GT) than a MX 5 or MR2. If you uprate the engine to a turbo, then sensibly modify this with increased boost pressure etc. then I imagine that the Convertible would seriously worry some much more expensive machinary (including a normally aspirated 997). Which is exactly what I have in mind.
(ii) Personally, if someone gave me a 997, Murcielago etc. i would sell it as I wouldn't want the hassle of owning one, knowing how (in the UK anyway) you are an immediate target of envy and hate. I doubt that a subtly modified 300 ZX would attract anything worse than admiring glances and the odd traffic lights Grand Prix.
(iii) Assuming that the base vehicle is sound and not too cosmetically awful. it seems to me that spending 5000 pounds or so converting a NA Convertible to a Twin Turbo, so that the whole lot costs you around 11,000 pounds, isn't such a bad deal. What else will you get for this money ? A tired-out Porsche 944 Turbo Convertible or mega-mileage 968 one ? A bottom-of-the-price-range Chimaera or Griffith, with big question marks over longevity ?
OK, maybe I am am justifying a semi-made decison, but (assuming that I can find the right car and someone to do the work to a high standard without ripping me off) the 300 ZX with a TT swapped for a NA sounds pretty good to me.