The AP ammo was heavier than the standard ball ammo, too. Not much, but enough to help it out at longer ranges??
The most accurate ammo I every ran through my 03A4 was my reloads using Sierra 180 grain match kings with IMR4350 powder.
Heavier bullets should be more accurate at longer ranges. They'll make it kick your shoulder harder, too. Learn to use that sling to help control recoil. Maybe put a rubber pull over kickpad on it if you can find one that will fit the stock - or wear a coat/jacket with a padded shoulder. Seriously, my 03A4 is way more noticeable to me than my M1 Garand (and people say those kick hard).
While the issue scope makes it look correct, you'd be better off with a more modern scope with bigger lens, better clarity, more light transmission, etc.