The rear sight looks original (although the "7"s on the leaf should have straight bases). The AV barrel MAY be from a post-WWI overhaul.

A grasping groove stock would be correct. A 1909 or 1910 RIA barrel would also be correct, but they are pricy and hard to find.

My feeling, is that most of the parts looks "period" correct (post WWI) and I would be tempted to keep as many as possible, including the barrel. Rust can be cleaned off. The rear sight really can't be improved on, as far as looks. Considering it is a low numbered M1903, I don't think it is worth going to extraordinary means to update.

I would look for a stock, upper band and lower band and leave it at that.

Look on the top and bottom of the bolt and on all sides of the safety lug (about halfway up the bolt body) - what markings do you see?