The social welfare of a unique general competitive equilibrium in an exchange economy
Using the 'utility monster' critique of aggregate utility maximization it is pretty easy to show that gen. comp. eq. does not always maximize aggregate utility.

Imagine a pure exchange economy with two agents, $A$ and $B$ and two types of goods, $x_1$ and $x_2$. Agent $A$ has nothing whereas agent $B$ has one unit of each good. The agents respective utility functions are
$$
U_A(x_1^A,x_2^A) = 2x_1^A + 2x_2^A \hskip 20pt U_B(x_1^B,x_2^B) = x_1^B + x_2^B.
$$
Trivially the initial distribution is a competitive equilibrium and also trivially it does not maximize aggregate utility.



EDIT:

I am not sure I understand @spinkus's comment, as obviously not every solution maximizes the aggregate utility, but I will address the other point: Not trading. That is not at all important for me, I just wanted to make the example trivial.

Suppose given some strictly monotonic utility functions $U_A,U_B$ and some initial endowments $w_A,w_B$ some allocation $a^*$ is a competitive equilibrium. If this allocation does not maximize aggregate utility we are done. If it does, consider the following monotonic transformation of $U_B$:
$$
\hat{U}_B \triangleq c \cdot U_B
$$ where $c$ is a positive number. In a pure exchange economy with $U_A,\hat{U}_B$, initial endowments $w_A,w_B$ the competitive equilibrium will again be $a^*$ as the preferences are unchanged. Yet even if $a^*$ maximized $U_A+U_B$, it clearly will not maximize $U_A + \hat{U}_B$ if $c$ is small enough.