This is a different type of decision problem, it doesn't match what is usually called "bounded error complexity". 

In BQP and similar classes we decide if an explicit input $x$ (that has a precise mathematical description) is in a well-defined language $L$, i.e. in some particular set of input instances. 

We can give a description of $L$ that uses access to oracles, but we must be sure that oracles act like precise mathematical functions. So that a definite input $x$ leads to a definite output $y$, which then certifies if $x$ is in $L$ or not. Our task is to decide efficiently if $x$ is in $L$ or not. 

For example, in Grover's algorithm the input $x$ is from a set $\{0,1,\dots,N\}$ and the output is $1$ or $0$ that we know through a quantum oracle $U_f$.
The language $L$ is the preimage of $1$.

But in your case the input **is the unknown oracle** $U$, in essence. And we want to figure out its property, either $U|0\rangle$ is entangled or not. 

This problem requires a careful modification, so that one would be able to speak about the complexity of a solution.