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.

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.}