It won't be a problem at the UK airport to go out but for coming back, expired EEA Family Permit wouldn't work. You will have to renew it before you travel to UK otherwise the check-in staff wouldn't even let you board the plane.
The advise of renewing it in Italy because you are travelling there seems adequate but whether UK embassy in Italy would renew it for you or not is a different story. I guess you are going to Italy for a short trip only which can explain why would the embassy in Italy refuse to help. You normally need to be resident in the country if you expect the embassy in there to renew your permit.
Hope this helps