How to add a lamp on the other side of the switch series?
Be aware that any voltage drop across the relay coil will make LS1 correspondingly dimmer, so choose the correct type of relay. For this series-wired application you want a current-actuated relay rather than a voltage-actuated type. 

A simple current-actuated relay has only a few turns in its coil, which is wound with wire sized to handle a lot of current. For your application the winding might have one to three turns, and the wire must be at least as large as your household wiring. 

When you shop for the relay, you may find it easier to buy an electronic device than a simple relay. This would probably operate on the same principles as a GFCI. Its voltage drop and power consumption will be too small to measure. Such a device may require its own constant-hot and neutral connections, which fortunately you have available at S2. 

One disadvantage with this circuit is that if lamp LS1 burns out or is unscrewed, lamp LS2 cannot be switched on. - Nevertheless I believe you have devised the best solution for the constraints as described.