Set of Bayes Correlated Equilibria when complete information is not available
You can certainly do it. However, keep in mind that the BCE will not be a lot "smaller". This is because there are many information structures that are almost perfectly informative, or that fully reveal some states but not all of them. Therefore, by disallowing the complete information structure you are simply removing one element in the boundary of the set of BCE's. Keep also in mind that this set has, in general, infinitely many elements. Since it is a convex set, any convex combination of two elements in it is also in the set, for example. 

Going back to your question, there are many ways to rule out complete information structures. The one that comes to mind stems from the fact that a fully informative signal will correspond to a BCE where for each $y\in Y$, the support of $P_{V|Y}$ is a singleton. Notice that if this is the case, learning what action you should take implies learning the state of the world. 

Therefore,  the extra constraint would be that $P_{Y,V}$ must satisfy that there exists a $y^*\in Y$ such that the cardinality of the support of the conditional probability of the states given that action is bigger than 1, i.e. $|P_{V|y^*}|&gt;1$. This is consistent with the idea that at least in some cases the DM remains uncertain about the state of the world after receiving the recommendation to choose action $y^*$.

Notice that a signal that fully reveals all but one state of the world, and instead induces a posterior belief for such a state of $99.999\%$ will not be ruled out by the extra constraint, so including the constraint will have very little effect.