M1903A3 is a good choice if you plan to convert it into an M1903A4 sniper rifle; otherwise get an M1903.
Also, be mindful of the serial numbers; you want to have a "high Serial Numbered" rifle (800,000 or higher on SA rifles; 286,506 on Rock Island Armory rifles) because the "low" serial numbers were forged a single heat treatment and tended to shattered, blasting the bolt rearwards into the shooters face.
After several incidents, the forging technique was changed and a "Double Heat Treatment" was done on all rifle receivers with a SN of 800,000 and higher on receivers made by Springfield Armory and 286,506 on receivers made by Rock Island Armory, majking these rifles safer to shoot.
My personal M1903A1 is a high SN (975,XXX) Springfield Armory receiver and I have used commercial ammunition (150gr. FMJBT loads) without a problem.
I suggest you go to www.m1903.com and read up on the information regarding receiver failures on the M1903 Springfield; very valuable information.