The fixed point set of the involution in the universal cover is a totally geodesic copy of hyperbolic space of codimension 1.  So that gives you the first totally geodesic hypersurface in some finite cover of the given thing.  And since the hypersurface is the same type of lattice, there is a finite cover of it that has a totally geodesic hypersurface by the same argument.  There is a single finite cover of the original thing doing both at once (first a cover to get the first hypersurface embedded, then another to get the second embedded).