If it is a 2013 model then it is still under warranty. The Leisure battery is Original Equipment, so if removed by Dealer or original Purchaser then this might well invalidate the Warranty. I would ask the seller, after giving him the info where the battery should be ie: under the front passenger seat, to confirm in writing that there is / is not a Leisure Battery fitted and if not ask him for a letter confirming that the VW warranty is complete and in force for the change in wiring specs. He can do this very simply by going back to the Dealer he purchased it from.
I would think the battery is present and that the original owner is a bit of a numpty. If the battery is present and flat because of a problem with the charging circuit ( you said the back lights can only be used with the engine on) then VW warranty should sort this out.