There should be no movement between the frame and barrel for best accuracy, if there is, it will affect accuracy. .001" movement at the breech end of the barrel can affect bullet POI about 12" at 100yds, so it's imparative for accuracy that lock up be identical each time the action is closed, if there's movement, there's a good chance it won't be locked up the same each time. Barrel fitting instructions in the FAQ cover how to fix it, H&R would replace the barrel if it was factory fitted.
Bedding the barrel in the frame would dedicate the frame to the .223 barrel, the 308 barrel most likely wouldn't fit. You'll most likely find you have a ~.001" gap at the chamber end of the barrel, a good fit will be tight on a .001" blade.
Tim