@DAPPER What I'm suggesting is that you plug one of your cables into the socket on an extender. It simply will be acting as a receiver then putting the signal down the wire to the outbuilding. People use extenders in this way to connect to devices that don't have WiFi.
If you have a suitable extender with an ethernet outlet lying around it should be relatively easy to try this with some extra ethernet cables and sockets at each end and a suitable device with an ethernet input to try in the outbuilding.