There seems to be a classification of representations for the example you mention (and more generally for representations of $C_p$ over $\mathbb{Z}/p^s\mathbb{Z}$) in

V. S. Drobotenko, E. S. Drobotenko, Z. P. Zhilinskaya and E. V. Pogorilyak,
Representations of cyclic groups of prime order $p$ over rings of residue classes mod $p^s$, Ukrain. Mat. Z. 17 (1965), 28-42. MR0188304.

I've not been able to find a copy of the paper, which is in Russian, but if I understand correctly the review in Math Reviews, it seems as though the indecomposable representations, up to isomorphism, are as follows:

- For each monic irreducible polynomial $\varphi(x)$ over $\mathbb{Z}/2\mathbb{Z}$, and each $e\geq1$, a representation sending the generator of $C_2$ to $I+2X$, where $X$ is a companion matrix of $\varphi(x)^e$.
- For each monic irreducible polynomial $\varphi(x)$ over $\mathbb{Z}/2\mathbb{Z}$, and each $e\geq1$, a representation sending the generator of $C_2$ to $-I+2X$, where $X$ is a companion matrix of $\varphi(x)^e$.
- The regular representation.

These are all non-isomorphic except that the rank one representations given by (1) and (2) are the same.

The representations that are irreducible (in the sense of having no $\mathbb{Z}/4\mathbb{Z}$-free subrepresentations) are those that occur in (1) and (2) for $e=1$. Perhaps $K_0(G,R)$ will be $\mathbb{Z}$-free on the classes of these?

**Edit:** I don't think the statement of the classification can be quite right, as it seems to me that the representations given in (1) are isomorphic to those given in (2):

Let $X$ be the companion matrix, over $\mathbb{Z}/2\mathbb{Z}$, of a polynomial $p(t)$ and let $Y$ be the companion matrix of $p(t+1)$. Then $I+X$ has minimal and characteristic polynomial $p(t+1)$ and so is similar to $Y$. So $(I+X)A=AY$ for some invertible matrix $A$.

Lift $A$ to a matrix $\tilde{A}$ over $\mathbb{Z}/4\mathbb{Z}$. Then $\tilde{A}$ is automatically invertible, and $(I+2X)\tilde{A}=\tilde{A}(-I+2Y)$, so $\tilde{A}$ gives an isomorphism between the representation where a generator of $C_2$ acts as $I+2X$ and the one where it acts as $-I+2Y$.