  We calculate accurate potential energy curves for a ground-state He$^+$ ion
interacting with a He atom in the lowest-energy metastable $^3\!S$ electronic
state. We employ the full configuration interaction method, equivalent to exact
diagonalization, with results extrapolated to the complete basis set limit. The
leading relativistic and adiabatic corrections are included using perturbation
theory. We calculate rovibrational levels and spectroscopic constants of the
He$_2^+$ molecular ion in excited electronic states for three stable
isotopologues. We predict the scattering lengths for ultracold ion-atom
collisions. The theoretical data are presented with their uncertainties and
agree well with previous results for the ground state. The reported results may
be useful for the spectroscopy of the He$_2^+$ molecular ion in the excited
electronic state and collisional studies of He$^+$ ions immersed in ultracold
gases of metastable He atoms.
