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"...heavy or light bullets for short distance..." Distance doesn't matter much. Use any 165 grain hunting bullet for the '06. (Partial to IMR4064 myself.) Sight in about 3.5" high at 100 and you'll be on target out to roughly 300 yards with no hold over. Same powder and sight-in for a 168 grain match bullet(out to 600 yards). Assuming you're just target shooting at 100, a 150 or 155 grain match bullet, also with IMR4064, will have a bit less felt recoil.
"...they take longer to stabilize..." Takes any .30-06 or .308 bullet 300 yards to stabilize. 300 yards is short range for NRA (and Canadian DCRA) High Power match shooting.
Don't shoot a 6.5 or .223 myself. However, the rifling twist on the AR says heavy bullets. That'd be 69 plus grains.
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