The biggest problem is fitting engine with transmission, but I've seen many different engines matched with many different gearboxes. The problem is that you don't have many choices for swapping transmission. You can use original or one from Fiat 126, VW Beetle or similar VWs, Porsche 911. I've seen many Beetles with VW diesel engines fitted. That can be done easily, especially when you're using LT3 (Transporter) gearbox.
Of course you can use any engines mentioned above by elhigh. Or Smart's CDI engine, that would make your Fiat flying
Anyway I think that you can rebuild your gas engine with good impact on FE. There are some mods that you can make to increase it's efficiency. And it's always nice to have a "stock" car. But I've made some strange swaps on my own, so I get your idea