This answer address both the original question and the poignant comment by YCor.
Such a concept of an "enriched" Grothendieck ring mapping onto $K_0(V_k)$ has already been formalized, pun intended, by H. Schoutens around 2009–2012 in two preprints, found found here: Schemic Grothendieck rings and motivic rationality and The yoga of schemic Grothendieck rings, a topos-theoretical approach.
One of the main innovative ideas here is to use the functor of points perspective of Algebraic Geometry to pick up a non-empty compliment $X^{\circ} \setminus X_\text{red}^{\circ}$ where $Y^{\circ}$ denotes the functor of points from finitely generated $k$-algebras to sets given by a separated scheme of finite type over a field $k$.
One can define several interesting motivic sites on the category of separated schemes of finite type over a field as different natural collections of sieves which happen to form a distributive lattices (so that scissor relations hold and fiber products work as multiplication).
The most interesting of these sites is the formal site $\mathbf{Form}_k$ whose Grothendieck ring I will denote by $\mathbf{H}$. It is the right concept in my view as it takes into account thickenings of arbitrary order. Of note, in $\mathbf{H}$,
$$[\mathbb{P}_k^1] = \mathbb{L}+ \widehat{\mathbb{L}} $$
where $\widehat{\mathbb{L}}$ denotes the class of the formal sieve given by the functor of points of the completed affine line at the origin.
Note further that there is a natural surjective ring homomorphism $\mathbf{H} \to K_0(\mathbf{Var}_k)$ given by sending the class of a formal sieve $[\mathcal{X}]$ to $[\mathcal{X}(k)]$ and if you complete these rings along the dimensional filtration, you obtain continuous ring homomorphisms of topological rings (cf., Theorem 2.1 and Lemma 2.4 of my paper here: On the auto Igusa-zeta function of an algebraic curve).
Note then that it is immediate that $[X^{\circ}]$ is sent to $[X_\text{red}]$. It is slightly less immediate but true that $[(\widehat{X}_Y)^{\circ}]$ is sent to $[Y_\text{red}]$ so that in particular $\widehat{\mathbb{L}} \mapsto 1$ under this ring homomorphism.
A few things I find interesting about this ring:
- You get to localize by a bunch of stuff for free since many things get mapped to 1.
- Relatedly, it contains a highly nontrivial subring $\mathbf{H}_0$ formed by all zero dimensional formal motifs which gets mapped to $\mathbb{Z} \subset K_0(\mathbf{Var}_k)$.
- It is very rigid: I suspect that $[X^{\circ}] = \mathbb{L}^n \implies X \cong \mathbb{A}_k^n$.
- The possibility of some type of rationality of the motivic generating series of the form: $P_m : = \sum_{n=0}^{\infty} [(\underline{\operatorname{Hom}}(D_n, D_m))^{\circ}]\mathbb{L}^{-r_n}t^n$ where $D_i : = \operatorname{spec}(k[t]/t^{i+1})$ and $r_n = n - \lceil \frac{n}{m}\rceil+1$ such that $P_m \mapsto (1-t)^{-1}$.
- Relatedly, the possible rationality of $A_n := \sum_{n=0}^{\infty} [(\underline{\operatorname{Hom}}(D_n, D_n))^{\circ}]\mathbb{L}^{-n}t^n$.