The $\mathbb{E}_\infty$ structure on the group completion corresponds to delooping the space in Nardin's comment to a spectrum. There is a unique homotopy class of spectrum maps $H\mathbb{Z} \to \tau_{\leq 1} S$ inducing multiplication by 2 on $\pi_0$, I would think the relevant spectrum fits in a pullback of the form $H\mathbb{Z} \to \tau_{\leq 1} S \leftarrow S$.

@ConnorMalin I think $S^1$ and $[0,1]$ behave quite differently: for instance, any homotopy equivalence becomes a simple homotopy equivalence after crossing with $S^1$, see Theorem 23.2 in A Course in Simple Homotopy Theory by M. Cohen.

The previous comment was about smooth manifolds and diffeomorphism, where the map to $BO$ classifies the stable tangent bundle. For topological manifolds and homeomorphism, replace $BO$ with $BTOP$.

In between the two abelian semigroups in the question, one could consider $\mathcal{M}an^{th}$ defined as manifolds up to homotopy equivalence over $BO$. Maybe Proposition 3.3 of this paper can be used to show that $K(\mathcal{M}an) \to K(\mathcal{M}an^{th})$ is an isomorphism? I'm not sure how to proceed from there though, but at least it would be "purely homotopy theory".

Is the characterization theorem stated correctly here? The version of Sakai's book that I could find requires only that $\bar{f}$ be defined on a neighborhood of $B \subset A$. It also has an additional condition, namely that $X$ be the direct limit of a sequence of finite-dimensional compacta.

Interesting answer! Unimportant for the conclusion, but I don't understand the sentence "This could be dropped since $X$ is connective".