Information and search
The solution concept of Bayes Correlated Equilibrium applies to games, viz strategic interactions, between multiple players. Thus, in a single person decision problem its use seems to me inappropriate or at least superfluous. A lot of effort has been spent over the last 70 years (dating back at least to Blackwell 1951, 1953) to explore the notion of information and information structures in decision problems. In that literature lies your answer.

What you have described is a search problem that seems analogous to a Wald-like sequential sampling problem. 

Perhaps refer also to the recently burgeoning literature on rational inattention. There, care has been taken in order to explore so-called "posterior separable" cost (of information) functions, which allows one to write the cost of information just as a function of the induced posteriors. See recent work by Caplin, Dean, and others.