We study a setting in which an active meta-learner aims to separate the idiosyncracies of a particular task environment from information that will transfer between task environments. In a Bayesian setting, this is accomplished by leveraging a prior distribution on the amount of transferable and task-specific information an observation will yield, inducing a large dependency on this prior when data is scarce or environments change frequently. However, a misspecified prior can lead to bias in the inferences made on the basis of the resulting posterior --- i.e., to the acquisition of non-transferable information. For an active meta-learner, this poses a dilemma: should they seek transferable information on the basis of their possibly misspecified prior beliefs, or task-specific information that enables better identification of the current task environment? Using the framework of Bayesian experimental design, we develop a novel diagnostic to detect the risk of non-transferable information acquisition, and leverage this diagnostic to propose an intuitive yet principled way to navigate the meta-learning dilemma --- namely, seek task-specific information when there is risk of non-transferable information acquisition, and transferable information otherwise. We provide a proof-of-concept of our approach in the context of an experiment with synthetic participants.