My first impression is that yes, the circuit would complete. Keep in mind that the metal has both negatively charged electrons which are reasonably free to mov

My first impression is that yes, the circuit would complete. Keep in mind that the metal has both negatively charged electrons which are reasonably free to move around within the metal and positively charged protons that are generally in the nuclei which are fairly fixed within the metal. If you have the metal belt moving so that the electrons are not actually moving (from the point of view of a stationary observer outside the belt) that observer would see the protons moving. You've still got, from the outside observer's point of view, a current. It just is now a proton current instead of an electron current.