It's a very late M1903, truthfully.
And falls in the serial range of what were called M1903(M), for Modified, as produced by Remington using M1903 parts as they tooled up to produce the M1903A3 version during the war. Why the barrel is off by two years is a matter of conjecture - it could've been rearsenaled at a later date, or perhaps the receiver sat for a while prior to final assembly. Regardless, the barrel may indeed be a 1903A3 barrel, but the 1903 rear sight sleeve has already been installed, making it for all intents and purposes a 1903 barrel. (Minor differences in the front sight fitment between 1903 and 1903A3)
The Remington M1903 (M) rifles turn up from time to time, using the search function here I've seen at least a couple others with serial numbers very close to yours. Note that some historians argue that there was really no 1903 (M) variant, because Remington was in a constant state of product improvement as they transitioned from 1903 to 1903A3 production. So you'll see a variety of different parts and pieces on the 1903 (M) variant as Remington sought to streamline their production capabilities. Discussion of that here:
http://www.jouster.com/cgi-bin/03/03config.pl?noframes;read=37965
The Springfield Historical Society website lists specimens by serial number and their disposition. While your rifle isn't in there (you have to add it to their database) one rifle fairly close to yours in serial range shows issue to the Lake Erie American Legion post.
At any rate, it's a nice specimen of a transitional M1903 Springfield as made by Remington early in WWII, just prior to the changeover to M1903A3 production. I'd leave it in exactly the condition you found it, because I'd wager there's some historical significance as to why that receiver has a 1944 barrel installed. It won't hurt to take the action out of the stock and take a look at the barrel sleeve/barrel junction, particularly the metal finish, and see if you can tell whether it was a later barrel retrofit or left Remington that way.
