Yeah, the Remington Hevi-Shot loads were terrible about spewing buffer too. After shooting a couple of boxes of them, I started apply my own seal to eliminate the mess. Ah, the good ol' days. Those original Remington HS loads were all the rage for awhile...
As to the OP's question, yes, copper-plated shot should penetrate a bit better, but it's not gonna turn it into hevi-shot either. I'd prolly run the lead if it's patterning significantly better or the copper-plated if the lead is patterning only marginally better.