Here's the definitions I'm using, just in case.
Let $E$ be the exchange economy given by agents $A,B$, starting allocations $x_A=(0,1)$, $x_B=(1,0)$ and utility functions given by $u_A=x+y$ and $u_B=x+\sqrt{y}$. It's easy to check that the core of this economy is comprised of any allocation $x_A=(t, 1/4)$, where $t\in[3/4, 1-\sqrt{3/4}]$ and that the only competitive equilibrium is the one given by $x_A=(3/4,1/4)$ together with the price vector $v=(1,1)$.
How can I prove that for every non-competitive core allocation $x$, the 2-fold replica of $x$ is not in the core of the 2-fold replica of $E$?