Very much doubt if it would be a straight swap. Some sort of "virginisation" of a second hand unit would probably be needed? and then it would need to be electronically introduced to the vehicle systems? However let's take a wee step back before we go spending lots of dosh. There are two very common problems on small Fiats which need checked out first as they can cause all manner of problems.
This is going to sound stupidly simple but get your battery properly checked out. It needs to be in tip top condition, these vehicles don't like "iffy" batteries and just because it starts the engine doesn't mean it's good enough to satisfy the electronics! Also take a detailed look at the engine earth cable which can corrode badly. Both these things can cause stop/start to drop out and other "strange" things to happen.
If the battery is good then you need to identify exactly what the car thinks is wrong with it so you need to get a read out of stored trouble codes and you'll need a code reader or scanner for that. If you plan on working on Italian cars much then I can strongly recommend
Multiecuscan:
https://www.gendan.co.uk/multiecusc...8DtJ8kna_zbH6Y3SZU2hYMJ3qT7u9sdkaAkcKEALw_wcB I've intentionally linked you with Gendan who are well worth a call to discuss what might be best for you. I've bought stuff from them, including Multecuscan and find their backup second to none.
Hopefully once you've scanned the modules for codes you'll be much better informed about what the way forward might be - which may, or may not, include installing a new Body Control Module.
I have a flashing mileage indicator on my boy's 2012 Punto at this time - a not uncommon symptom I believe. I haven't yet had a chance to scan it for codes but I suspect a communication problem involving the BCM and I may be able to sort it by running a Proxi Alignment using
Multiecuscan. This will be very exciting for me as I've never done a proxi alignment before! Here's a video which shows it being done on a 500, I believe it's exactly the same procedure on his Punto: