The most concrete way of constructing Kirchberg algebras with given $K$-theory that I know of, is due to Rordam and Elliott-Rordam, and is given in a series of papers "Classification of Certain Infinite Simple C*-Algebras" (Rordam) and "Classification of Certain Infinite Simple C*-Algebras, II" (Elliott-Rordam). There, Kirchberg algebras are constructed as crossed products of either AF-algebras or AT-algebras (depending on whether there is torsion on $K_1$ or not) by endomorphisms, similar to how the Cuntz algebras are obtained from UHF-algebras. I suppose one could go through their construction and see how they get the AF/AT-algebras and the endomorphism from the given $K$-theoretical data (which in general will also include the class of the unit), and then try to see if there is any way of relating the algebras one gets for two different choices, and the algebra one gets for the sum of the $K$-groups. One complication is that all the AF and AT-algebras involved are simple (so taking the sum of the AF/AT-algebras won't work either).