Greetings to the forum. There is a wealth of info on here, and even more info that hasn't been posted yet.
So regarding the engine. First thing we should do is determine what engine the car has in it right now. Its not uncommon for these cars to have their engines swapped and changed about. Most stock 500's came with the standard 499cc (110D/110F engine designation). early ones were 479cc(110 designation). There were also special Abarth blocks that were 594cc..
With that said, the fiat 126 shares nearly an identical engine layout as to the 500, with minor modifications to the starter location. Upgrading to a 650cc 126 engine will require the 126 mating transmission, or bell housing with the alternate starter location.
heres some of the block designations for 126 based engines.
126A5 000 - from 594cc fiat 500R's
126A 000 - from 594cc fiat 126's
126A1 000 - from 652cc fiat 126's
126A2 000 - from 698cc fiat 126 BIS engines.
The engine designation is stamped on the front case, right beside the fuel pump. If you snap a photo of it, we can then help with routes to go. This will help determine how mild or wild you can go without getting into too much machining work.
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Typically the cheapest and most pain free route to go is swap in a 650/126 engine combo.. However, that's not always the easiest route on this side of the world. Finding those engines and transmissions is not always easy. Importing one can also be hit and miss regarding its condition.
Do you plan to do most of the work yourself (minus any machining work to be done if needed)?
On a side note, my car (currently being restored), has a 540cc engine with some goodies thrown to it (soon to be dyno'ed). My 2nd engine, slowly being built will be a monster 866cc with some significant modifications and machining done. You can go mild to wild easily with these cars lol.