Based on the information you provided I believe that your theory that he was in the 82nd Airborne is correct.The 101st did not serve in Italy.
As to the 82nd, that is something of a wild card.The primary maneuver elements of the 82nd Airborne were the 504th and 505th Parachute Infantry Regiments; 325th Glider Infantry; 376 Parachute Field Artillery Battalion and the 307th Engineers.All of these units served in Italy.
The 504th did not participate in Normandy except for a small contingent of 26 men who jumped as Pathfinders.All of the other units mentioned did participate in Normandy so would have a FRANCE credit in their record.
As to the 325th Glider Regiment, many of its men were cross trained as paratroopers so that is a marginal possibility.
The 82nd was reenforced by the 507th and 508th Parachute Regiments for Normandy, but these units did not have an ITALY history.
So the only manuever elements of the 82nd that provide strong possibilities are the 307th, 376th and the 505th.
As far as Italy goes, you mention that his dates in theater are 1944-45.The 82nd jumped in Sicily in July 1943, and then on the Italian mainland in September 43.The bulk of the 82nd was withdrawn from the Italian Campaign in Feb44 and returned to England to prepare for Normandy, leaving the 504th in theatre until late April44.
So the 505/376/307 gets a credit for Normandy (France) and the entire Division also got a campaign credit for Germany in early 1945.However, they also participated in the Invasion of Holland (SEP44).And the Battle of the Bulge DEC44-JAN45 (Belgium/Germany)
If he was with the 82nd for its full tour he would have four stars tacked to his jump wings.The 82nd is credited with four combat jumps in WW2: Sicily-Salerno-Normandy-Holland.The 43 jumps you reference are most likely a cumulative number including training, practice and combat jumps.
Hope this helps some.Feel free to drop a line if any clarification is needed.