Non-invertible version of unitary intertwiners between correspondences of $C^\ast$-algebras There is a version of the Eilenberg-Watts theorem for $C^\ast$-algebras, where functors between appropriate module categories correspond to what are called 'correspondences' of $C^\ast$-algebras. These are one-sided Hilbert modules for one of the $C^\ast$-categories together with an action on this module by the other $C^\ast$-algebra. In the literature one finds a (weak) (2,1)-category where the objects are $C^\ast$-algebras, 1-arrows are correspondences, and 2-arrows are unitary intertwiners. However, clearly if one thinks of the categories of representations, the appropriate functors between such categories, and the natural transformations between these functors, then one could have non-invertible 2-arrows. 

What intertwiners play the analogue of the non-invertible natural transformations?

 A: As you note in your comment, a lot depends on which category of modules you choose to consider. 
If you work with the category of Hilbert $C^*$-modules, with adjointable operators as morphisms, then the appropriate kind of maps between correspondences would be the adjointable bimodule maps: for $C^*$-algebras $A$ and $B$, the category of (nondegenerate) $C^*$-correspondences from $A$ to $B$, with adjointable bimodule maps as morphisms, is equivalent to the category of strongly continuous $*$-functors from $C^*$-modules over $A$ to $C^*$-modules over $B$, with natural transformations as morphisms. This is due to Blecher (Mathematische Annalen, 1997). 
If, on the other hand, you consider operator modules, as in the paper of Blecher cited in the comments, then the appropriate class of morphisms would be completely bounded bimodule maps---and, one could argue, the appropriate class of bimodules is that of operator bimodules. In the paper you cited (Math. Scand. 2001), Blecher proves the remarkable fact that every invertible operator bimodule between $C^*$-algebras is a $C^*$-correspondence.
