You might want to check the twist rate in your rifle. A lot of 243s are 1:10 which is great for weights under 90 grains. My Model 70 is a one-hole wonder with 87s at 100 yards. Using 100gr bullets, I'm lucky to get 4 MOA.
Don't sweat not being able to use a heavy bullet, if your rifle doesn't like them. A 70 Nosler ballistic tip and a number of other choices in under 90gr selection are devastating on medium-sized game -- especially hits in a vital area.