I am no expert on this. Late 1944 is the timeframe a couple sources have given for Allied re-equipping Italianicon forces. This entirely makes sense when looking at the timeline and organizations.

Shortly after Marshal Badoglio took over, secret negotiations about surrender to the Allies began. As you may know, the Armistice of Cassibile became public a little off schedule (at least from the Italian point of view). Regardless, it probably would not have been enough to block Operation Achse, which effectively broke the Royal Italian Army. The remnants had to be rebuilt, which took time.

The first combat unit to be formed to fight with the Allies with the 1st Motorized Group (Nov 1943). By the Spring of 1944 this evolved into the Italian Liberation Corps, consisting of two Divisions of roughly 24,000 soldiers. But in Sept 1944 they were taken off the line and re-organized and requiped into a yet larger force called Italian Co-belligerent Army.

Operation Achse and the need for Allied forces to be armed with ammunition that can be resupplied make it farily apparent why they would be re-equipped with Allied weapons. How many of those where US small arms, and whether any carbines were issued at that time, I don't know. All I'm saying is that it's possible.