Dear Bas,
It seems hard to find references to a categorical treatment of C*-algebras.
I am surprised to hear that. Here are a few, which I am almost sure that you are already familiar with: D. H. Van Osdol. C*-algebras and cohomology. In Categorical topology (Toledo, Ohio, 1983), volume 5 of Sigma Ser. Pure Math., pages 582-587. Heldermann, Berlin, 1984. J. Wick Pelletier and J. Rosick´y. On the equational theory of C*-algebras. Algebra Universalis, 30(2):275-284, 1993. Joan Wick Pelletier and Ji.r´. Rosick´y. Generating the equational theory of C*-algebras and related categories. In Categorical topology and its relation to analysis, algebra and combinatorics (Prague, 1988), pages 163-180. World Sci. Publishing, Teaneck, NJ, 1989. Edward G. Effros and Zhong-Jin Ruan. Operator spaces, volume 23 of London Mathematical Society Monographs. New Series. The Clarendon Press Oxford University Press, New York, 2000. [This last one is not categorical, but it contains some results concerning the tensor products that can easily be interpreted categorically.] You my find a brief summary of the categorically interesting points in Chapter 8 of my PhD thesis: http://at.yorku.ca/p/a/a/o/41.pdf
Concretely, there are several tensor products on C*-algebras. Which one is `the right one' from a categorical perspective?
This is an interesting question, but I suspect that you may find a clue to answer this question here: Theodore W. Palmer. Banach algebras and the general theory of *-algebras. Vol. 2, volume 79 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2001. I hope that my answers are of some help to you. Best wishes, Gabi