Given the area of the crash the fighter may have been in a unit supporting the drive of the 7th Army and French 1st Army from southern France. I assume the 12th Air Force supported the armies initially. (XII Tactical Air Command?) Later it was the 1st Tactical Air Force (Provisional) but I think it was organized in November 1944. Even though the crash was in September, perhaps we should look at the fighter groups that eventually were assigned to the 1st Tactical Air Force.
https://www.amazon.com/First-TACAF-T.../dp/1891570072