Re: 1903 sniper?
Obviously and positively not a Military or Arsenal built Sniper Rifle.
The action is a 03 Mk I receiver with the second version magazine cutoff. It was made to accept the pederson device to shoot .30 cal pistol ammo for training at lower cost, and in smaller safety zone ranges (like indoors). Good luck ever finding a Pederson device.
The Stock is a Raritan Arsenal (Edison NJ, 1917-1963) rebuild stock.
Marine Sniper and Match 03's, equipped with a Unertl 8X would have had a front sight, most likely the wide USMC Match version, and a larger front sight hood, not those low, stamped and pressed ones on most WW II ones like the Remington manufactured ones, or the Smiths Coronas.
Unertl rings were readily available. Not enough info on the scope. looks to be of lete 20's/30's vintage. I know what it isn't, but not what it is.
The receiver sight is likely a Lyman, and it must be pretty ineresting trying to use it with the scope, or vice versa.
Somebody may have bought a surplus rifle in the late 20's or 30's, cheap, since it's a Mk I, and put on the Lyman sight. Then added the scope later to the same rifle without removing the Lyman. Pulling the slide up and out would allow the scope to be used, but you can't use the Lyman with the scope on. A ral WTF was he thinking. I see he left the factory ladder on, making it a very tight fit for the scope. That should have been removed with either of those sight addons.
Seems to me maybe somebody wanted to shoot some NRA shoots back in the 30's, and tried to upgrade instead of buying a new rifle from DCM. It might be something from Sedgley also, and had work done later. I doubt it is a Bannerman buy.
Post some pictures of te front sight, the front end of the stock, and full of the scope.
I expect all parts are milled.
I hope he didn't pay too much for it, and that the scope is servicable.