How does the electrolyte in a battery work?

Cu wire is neutral, so what allows it to conduct exclusively electrons in an external circuit and not positive ions?

 

You're objecting to one "neutral" medium transporting charge carriers, and you have no objection to a second "neutral" medium transporting charge carriers. It's the same question. What physical processes are you interested in understanding?

 

Cu atoms are playing "hot potato" with electrons given a potential difference between the ends of the conductor that is sufficient that electrons can make the jump, or be tossed from one atom to the next. What are analogous structures/entities in an electrolyte solution?

 

Generally, they do not. Depends on the cathodic and anodic reactions in the cell. If electrode potentials are such that cations are being reduced, cations will be moving from the "cell anode" where they are constantly being produced either by oxidation of anode material, or components of the solution to the "cell cathode" where they are reduced. Any given volume increment of the solution remains electrically neutral as anions play "hot potato" with the cationic drift. Other cell and electrode reactions might be forming anions at the "cell cathode" and those anions move to the anode to be oxidized.

 

This is a good diagram of the flooded lead acid battery redox. The charge separation from the chemical reactions creates an electric field and the free ions in the electrolyte move to neutralize the field in response until the system reaches equilibrium.

http://ecee.colorado.edu/~ecen4517/materials/Battery.pdf