It's actually an interesting scenario as it seems you'd need to comply with both sets of requirements to enter Italy - i.e. those applicable to having been in the USA and then those for having transited via the UK.

To comply with both sets of regulations you'd need to take your test on or after 10.50 am CET on 15th December (48 hours before your scheduled arrival into Italy) which would be 4.50 am EST in New York.

It still doesn't really answer the question whether AA would allow you to check in but you'd have time to take one in NYC between AA10's arrival and AA106's departure. If AA are happy to check you in just for the LAX - JFK sector then you'd be fine. You could test in NYC and then check in for JFK - LHR - MXP.

An alternative proposal would be to do the first half of your itinerary and then phone up to have the LHR - MXP sector dropped. That should see the itinerary repriced based on the the historic fares in play when you booked. Did you look at prices just for LAX - JFK - LHR on the return?