For a discrete abelian group $\pi$ the following simplicial abelian group models for $K(\pi,1)$ are isomorphic, I believe:

Simplicial abelian group whose normalized chain complex is $0\leftarrow \pi\leftarrow 0\leftarrow 0\leftarrow \dots$.

Nerve of category having one object and having morphisms $\pi$ (with obvious addition law since $\pi$ is abelian).

Universal example of simplicial abelian group equipped with a based map of simplicial sets from $S^1=\Delta^1/\partial$ to its underlying simplicial set.

When realized as a topological abelian group, this becomes the universal example of a topological abelian group equipped with a continuous based map from the topological space $S^1$ to its underlying simplicial set.

This latter space can also be described using configurations as in the previous answer.

In each of these constructions of $B\pi$, the universal cover $E\pi$ has its own corresponding description.

I believe that when $\pi$ has order two then $E\pi$ is homeomorphic to $S^\infty$, but I don't know if this abelian group structure on $S^\infty$ is the one referred to in the question.

$E\pi$ as a space, or even as a simplicial set, does not depend on the group structure in $\pi$; if you have a set $\pi$ then you can make $E\pi$ using the nerve of the category that has one object for each element of $\pi$ and one morphism between any two objects. If $\pi$ is nonempty then this is contractible, and if $\pi$ is a group then it gets an obvious free group action. It seems likely that if it is homeomorphic to $S^\infty$ when $\pi$ has two elements then the same is true when it has more than two (but finitely many) elements.

These don't seem to have the kind of strong minimality property that you are asking about: when $\pi$ has order two then there are plenty of contractible closed subgroups of $B\pi$ (for which the quotient is isomorphic to $B\pi$). But maybe that property is too much to wish for.

For a discrete abelian group $\pi$ the following simplicial abelian group models for $K(\pi,1)$ are isomorphic, I believe:

Simplicial abelian group whose normalized chain complex is $0\leftarrow \pi\leftarrow 0\leftarrow 0\leftarrow \dots$.

Nerve of category having one object and having morphisms $\pi$ (with obvious addition law since $\pi$ is abelian).

Universal example of simplicial abelian group $A$ equipped with a based map of simplicial sets from $S^1=\Delta^1/\partial$ to the simplicial set $Hom(\pi,A)$.

When realized as a topological abelian group, this becomes the universal example of a topological abelian group $A$ equipped with a continuous based map from the topological space $S^1$ to the space of homomorphisms from $\pi$ to $A$.

This latter space can also be described using configurations as in the previous answer.

In each of these constructions of $B\pi$, the universal cover $E\pi$ has its own corresponding description.

I believe that when $\pi$ has order two then $E\pi$ is homeomorphic to $S^\infty$, but I don't know if this abelian group structure on $S^\infty$ is the one referred to in the question.

$E\pi$ as a space, or even as a simplicial set, does not depend on the group structure in $\pi$; if you have a set $\pi$ then you can make $E\pi$ using the nerve of the category that has one object for each element of $\pi$ and one morphism between any two objects. If $\pi$ is nonempty then this is contractible, and if $\pi$ is a group then it gets an obvious free group action. It seems likely that if it is homeomorphic to $S^\infty$ when $\pi$ has two elements then the same is true when it has more than two (but finitely many) elements.

These don't seem to have the kind of strong minimality property that you are asking about: when $\pi$ has order two then there are plenty of contractible closed subgroups of $B\pi$ (for which the quotient is isomorphic to $B\pi$). But maybe that property is too much to wish for.