Nah, I wouldn't think so. I think the 1944 is more of a date that a serial number, assembly number or anything of the like. I'm not sure on the Italian proof facilities, but I'm highly confident that the German proof facilities were closed in the Summer of 1945 and reopened in September or October of 1945. I've read accounts that the Russian occupants under Major General Ivan Sazonovicha Kolesnichenko, Chief of the Soviet military administration in Thuringia, withheld black powder to prevent sabotage or an uprising. Either before the end of WWII or after October 1945, only black powder was available.
Right off, I can't recall seeing a German sporting weapon proofed in early 1945, but it is possible.

Below is one of the 1st Merkel(200E) to roll off the line and look whose initials are on the rifled tube; LK - Louis Kelber. Accounts give that Merkel was using surplus military blanks.


November 1945.

Kind Regards,

Raimey
rse