TITLE: K-theory of finite diagram categories
QUESTION [5 upvotes]: Suppose $I$ is a finite $\infty$-category and $F:I\rightarrow\text{fCW}$ is a functor that takes values in finite CW complexes. For each $X\in I$, let $[F(X)]$ denote the class of $F(X)$ in $K_0(\text{fCW})\cong\mathbb{Z}$.
Since evaluation at $X$ is a right exact functor, $[F(-)]$ is a homomorphism $K_0(\text{fCW}^I)\rightarrow\mathbb{Z}$. As we let $X$ vary over equivalence classes of objects in $I$, these assemble into a homomorphism $$K_0(\text{fCW}^I)\xrightarrow{ev}\bigoplus_{X\in I/\cong}\mathbb{Z}.$$ Question: Is this an isomorphism?
Remark: There is a splitting like this one when fCW is replaced by abelian/stable categories. See for example https://arxiv.org/abs/0908.3417
Edit: Tom Goodwillie points out this is not the isomorphism one would expect, but I am still curious whether $K_0(\text{fCW}^I)$ splits according to the objects of $I$ (and maybe their automorphism groups, as in the paper linked above), or whether it can be computed at all in general.
K-theory can mean so many things, I want to be clear: When I write $K_0(\mathcal{C})$, I mean the abelian group supporting a universal function $[-]:\mathcal{C}\rightarrow K_0(\mathcal{C})$ such that:

if $X\cong Y$, then $[X]=[Y]$,
$[X\amalg Y]=[X]+[Y]$,
if $A\rightarrow B\rightarrow C$ is a cofiber sequence, $[B]=[A]+[C]$.

REPLY [2 votes]: If $I = \mathbb Z/2\mathbb Z$ counts as a finite $\infty$ category (Edit: it doesn't), then the answer is no.  The space $* \sqcup *$  admits both a free and a trivial $I$ action.  We have an additive map given by taking homology and pairing with the alternating representation $$X \mapsto \chi(H^*(X, \mathbb C), {\rm alt}) = \sum_{i} (-1)^i \dim (H^i(X, \mathbb C) \otimes{ \rm alt})^{\mathbb Z/2}$$  which distinguishes between the two actions on $* \sqcup *$.