According to the online database of Italian KIA and MIA (
http://www.difesa.it/Il_Ministro/ONO...istrativo.aspx), there was only one Francesco Bargagna lost during the war and he is listed on 23 May 1943.
There are five Mario Mancini (seven replies but two duplicates) and one was also on 23 May 1943.
So it seems your survivor is not one of them.
Sadly you can only search with names, not by date. Unit is not given, only the death place and burial place. In both cases above, it is "unknown", normal for MIA people.
From Russel reply, it was a night or dawn action (with a take off time at 0300 hrs). Brian gives the date of 22 May 1943, does that mean the night of 22 to 23 May 1943 (and so the early hours of the 23rd) ?