A duality is discussed for Lie group bundles vs. certain tensor categories with non-simple unit. A classification of such tensor categories is given in terms of principal bundles, and a cohomological invariant is assigned, representing the obstruction to perform an embedding into the category of vector bundles. For each of these categories, we construct a K-group coinciding with the Nistor-Troitsky gauge-equivariant K-theory when the above-mentioned obstruction vanishes. Such K-groups can be used to compute the K-theory of a certain class of C*-algebras.