Below is an image of three Forster .308 dies. and a small base die will reduce the case body diameter approximately .003 smaller in diameter. Meaning it does not just size the base of the case body smaller.
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Clean inside the die body with Hoppe's and then polish the die with a snug-fitting shotgun cleaning mop. Put the mop in a drill and use Flitz, J&B Bore paste, automotive rubbing compound, Mothers Mag and Wheel polish, any of the above, and polish the die.
If you are sizing once fired Lake City brass the case might have been fired in a machine gun with a "FAT" chamber. And you might need to size the case first in a standard FL .223 die that is larger in diameter.
If you wet tumbled the brass this increases friction when sizing. And if the die was not perfectly cleaned of old sizing lube OneShot will not work properly.
Lake City brass is harder than commercial brass and if it was fired in a machine gun sizing can be a headache.
Below are a few of my .223 dies, the two dies I use the most are the Forster and RCBS small base die, And the Lee collet die and Redding body dies just gather dust.
If Chesty Puller was here he would tell you to get a Forster FL die.