It is not completely true that the set of gauge equivalence classes of Maurer-Cartan elements of a dg Lie algebra is invariant under quasi-isomorphisms. Here is an example (which I learned from Professor Andrey Lazarev): Let $\mathbb L(x)$ be the graded free Lie algebra generated by an element $x$ of degree $-1$. Then $\mathbb L(x)$ is 2-dimensional generated by $x$ and $[x,x]$ (by the graded Jacobi identity it follows that $[x,[x,x]]=0$). Consider now the dg Lie algebra $L=(\mathbb L(x),d(x)=-1/2[x,x])$, which has trivial homology.
We have that $x$ is a Maurer-Cartan element of $L$, but since $L$ is trivial in degree zero, it has a trivial gauge group and consequently $x$ and $0$ are not gauge-equivalent. However, $L$ is quasi-isomorphic to the trivial dg Lie algebra, which has a single Maurer-Cartan element.
Regarding the question of the thread: I have written a note regarding a possible candidate for gauge groups for nilpotent $L_\infty$-algebras. If there are no mistakes there, it is given by the degree 0 part of the $L_\infty$-algebra with the underlying additive structure. In particular, it is not a generalization of the gauge group of a dg Lie algebra, but have the same underlying set.
People I've talked to about this are pretty skeptical, so read at your own risk :).
https://drive.google.com/file/d/1Ti08qm9LM7CnbKNtjpHb9jYcaQ__TJ-P/view?usp=sharinghttps://drive.google.com/file/d/1M8WmINGHXIRAktr6yNy5a_qCdrlsyiwB/view?usp=sharing