Summary: A complete dessin of type \(\{2^e,2^f\}\) is an orientable map with underlying graph being a complete bipartite graph \(\mathbf{K}_{2^e,2^f}\), which is said to be regular if all edges are equivalent under the group of color- and orientation-preserving automorphisms, and circular if the boundary cycle of each face is a circuit (a simple cycle). As one of a series papers towards a classification of complete circular regular dessins of type \(\{m,n\}\), this paper presents such a classification for the case \(\{m,n\}=\{2^e,2^f\}\), where \(e, f\) are positive integers and \(e\ge f\ge 2\). We note that the group \(G\) of color- and orientation-preserving automorphisms is a bicyclic 2-group of type \(\{2^e,2^f\}\), and our analysis splits naturally into two cases depending on whether the group \(G\) is metacyclic or not. In this paper, we deal with the case that \(G\) is metacyclic.