I'll list some things I believe I can do, any of which might I might not actually be able to, and then, if I'm right, I'll ask if there's any point to what I've done.
The category of pointed locally compact topological spaces is a symmetric monoidal category under the smash product $\wedge$. Its skeleton $T_*$ is a (large) semiring $(T_*,\vee,\wedge,\{0\},S^0)$ if we take the wedge sum to be $+$, the smash product to be $\cdot$, the point to be 0, and the 0-sphere to be 1.
The category of all topological spaces is also symmetric monoidal under the product $\times$, with coproduct $\amalg$. Taking $0 = \emptyset$ and $1 = \{0\}$, I can make another semiring $T$.
Fix a field $k$. Then graded $k$-vector spaces become a symmetric monoidal category under $\otimes_k$, and the skeleton $M$ of this category is a semiring $(M,\oplus,\otimes_k,0,k)$.
In general given a symmetric monoidal category with finite coproducts, if the monoidal product distributes over the coproduct, taking the skeleton gives a semiring.
The functor $\tilde H^*(-;k)$, reduced singular cohomology with field coefficients, is monoidal, and when we take the skeleton, becomes a semiring homomorphism $f \colon T \to M$.
The Grothendieck group functor $K\colon \mathrm{Mon} \to \mathrm{Grp}$ takes semirings to rings, because functors preserve diagrams, in particular for monoid objects. So I make my categories into rings $K(T_*)$, $K(T)$, and $K(M)$, the same way one does for the categories of vector bundles over a space or representations of a group.
I have a ring homomorphism $K(f)\colon K(T) \to K(M)$. (I would like to say I have something more, because of the induced pullback maps in cohomology, but I doubt they survive the $K$-equivalence relation.)
Does anything? What I've said seems like a natural thing to do, so I'm sure someone has thought of it before, but I imagine if it yielded anything informative, I'd have heard of it, so it probably turns out to be boring. What goes wrong? Why is this boring?
The usual answer with these things seems to be the Eilenberg–Mazur swindle: for instance, take $Y$ to be the infinite wedge sum of any pointed space $X$, then $X \vee Y \cong Y$; so if you apply the Grothendieck group construction to the additive monoid, you get $X = 0$.
The moral of the story is that we cannot have a abelian group in which infinite sums make sense. So we are forced to restrict our attention to "finite" objects, as in e.g. the construction of $K_0 (A)$ for a ring $A$.