OP, you didn't say which year or model your car is! This info is important in order to get relevant advice....
IF the car runs ok when warmed up, then I think it's unlikely to be an ignition problem, so you could maybe skip the ignition checks/adjustments for the moment.
You say your Mechanic has determined the problem is that the choke flap (butterfly) isn't closing properly - I'll assume given his age/experience, he's correctly diagnosed the problem and has run some checks to rule out an ignition system fault.
And given that you ask about wiring up a carb. fitted with an electrical choke unit, I'll assume your carb has either a manually operated choke (i.e. pull the knob adjacent to ignition switch), or a coolant temperature operated choke unit.
If so, there might be no need to fit a new carb. nor need to wire up an electrically operated choke.
If manual choke (i.e. pull knob adjacent to ignition switch to operate) - this can usually be adjusted or fixed without changing anything.
If you have a coolant temperature operated choke, first check the adjustment of this unit and also the 'fast-idle' adjustment.
If these are ok, we ((I worked in a Fiat dealership, back in the day) used to replace just the choke unit, not the entire carb. While these choke units, on their own, are not listed by the likes of Autoricambi and MrFiat in the U.S., some of these choke units are still available from carb. specialists e.g. :-
www.webcon.co.uk/shopdisplayproducts.asp?page=2&id=682 (perhaps there are similar carb. specialists in the U.S.?)
If you decide to try to obtain and fit just the choke unit, you'll need to know the carb. model, this is stamped on the side of the bottom mounting flange of the carb. - remove the air cleaner and if necessary, use an inspection light + mirror to see it. If it's a Weber carb. it'll be something like 32DHSA; 34DMS; 28/36DHSA; 32ADFA etc, possibly followed by some other numbers to identify which jets are fitted etc.. If it's a Solex carb. e.g. C34EIES5, you're probably in trouble as spare parts for Solex carbs are very difficult to find nowadays.
This choke unit can be easily changed, could even do it yourself (it's not difficult). If you decide to do it yourself, I, or others on this forum, can guide you through the procedure.
Al.