The problem you are facing is based on an idealization of wires as perfect conductors. $\Delta V=0$ is true only if the resistivity of the wires is zero.
You have two ways to resolve this apparent paradox: either you consider a limit approach to infinite conductivity, where the electric field in the conductor $E=j/\sigma$ approaches zero in the limit, or you consider the small but finite conductivity of real wires.
In this latter case the field E in the wires is not zero. It's just very small and the voltage drop across any portion of wire is negligible.
In both cases, attaching a wire to a battery directly will result in exceedingly high current densities (infinite, in the limit for $\sigma$ that goes to infinite.)
I am not considering the actual case of a superconductor where the field inside would be zero and only a surface current will be allowed.