Maybe I don't understand what you're asking, but it sounds to me as though you should be able to select the Line, pick on any of its grips, hit Enter/space to get into the Move command, and then either object-snap to the 3D location if you can, or else use the .XY option and snap to the appropriate point on the Line, then give it a Z value or snap to anything in the Block that is at the desired elevation -- that should allow you to snap to some location where it's not going to snap to the Line again.