Let $R$ be a ring. The $K$-theory of $(Mod(R)^{f.g.proj},\oplus)$ is obtained by first throwing out non-isomorphisms and then group completing. What happens if these steps are reversed?

That is, consider the symmetric monoidal $\infty$-category $C = (Mod(R)^{f.g.proj},\oplus)$ of finitely-generated projective $R$-modules under direct sum (of course, this is in fact an ordinary category). To compute $K(R)$, we first pass to the core $\iota(C)$ to get an $\infty$-groupoid (which in this case is a 1-groupoid) and then we group complete to get $K(R) = \iota(C)[C^{-1}]$.

I'm thinking of group completion as an $\infty$-categorical process: for a symmetric monoidal $\infty$-category $C$, the group completion $C[C^{-1}]$ is the free symmetric monoidal $\infty$-category on $C$ where every object has a monoidal inverse; this agrees with the usual notion for symmetric monoidal $\infty$-groupoids. So it makes sense to reverse these steps and consider $\iota(C[C^{-1}])$. So my question is:

**Question:**

If $C$ is a symmetric monoidal $\infty$-category, then do we have $\iota(C)[C^{-1}] = \iota(C[C^{-1}])$?

If not, does this at least hold in the case $C = Mod(R)^{f.g.proj}$, yielding an alternate description of $K(R)$?

**Motivation:** My motivation here is basically to explore how robust the definition of $K$-theory is to perturbation. For example, an alternate description of $K(R)$ that definitely does work is the following: $K(R) = \iota_\oplus(\iota(Mod(R))[R^{-1}])$: that is, take the groupoid of *all* $R$-modules and rather than group completing, monoidally invert the single object $R$ itself viewed as an $R$-module; finally, throw away all objects which are not monoidally invertible (note in particular that since we're not monoidally inverting *everything*, we've sidestepped the Eilenberg swindle). I like this description because you don't have to put the "finitely-generated projective" condition in by hand -- it falls out naturally just from knowing that that the symmetric monoidal category $(Mod(R),\oplus)$ has a distinguished object $R$. If the answer to my question is "yes", then one could simplify this further to $K(R) \overset{?}{=} \iota_\oplus(\iota(Mod(R)[R^{-1}]))$; then even the choice of the $\oplus$ monoidal structure becomes more "canonical" in the sense that it has a universal property in $Mod(R)$, unlike in $\iota(Mod(R))$. Other variants start to suggest themselves, too.