Damn fine spot there Davren. The 850 spider/coupe had (I think) the larger 903cc four cylinder but in the US I have something at the back of my mind that says they used a higher compression version of the smaller 843cc motor which I seem to remember had something to do with emissions regulations not applying to engines below a certain size. 50 CID rings a bell.
The Moretti is probably a better known version of a Carozzeria building their own body onto a FIAT chassis. The 903 engine carried on well after the 850 went to its grave and powered the 127 and even early versons of the Punto which was available over here into the late '80s.
As for using 850 parts to rebuild it, well I don't see why not, it does depend though on what level of originality you want. If, on a vehicle this age, emissions regs are largely irrelevant, you could use a 903 sourced in Europe, but body panels will be much harder to come by. As it carries chassis no. 00002 there is an implication that it has some historical importance that would require a rebuild with the original engine in place as that will increase the value. There's obviously no point in spending $20,000 on it only to find out it's only worth $2,000.
Cars like the FIAT Dino Coupe and Spider can be bought in top notch condition for £20,000 to £25,000. It uses basically the same mechanicals as the Ferrari Dino 206 and 246 but is worth 20% to 25% as much as Enzo's car while costing the same to maintain.
If you do rebuild it faithfully then if you're married your other half is going to have to get used to having bits of engine in the bath and transmission components in the kitchen sink.
Obtaining a value for a restored model would be a good first move, and you only need to look at used Camaros to see how important to value matching numbers are. Tried buying an original Z28 with matching numbers recently?