Nice set-up for sure .... I've never seen a Battery Eliminator either, but assume it must be some form of capacitor.
Who manufactured that sounder .... Wunel or something like that?
Are there any T&P markings on either the sounder or the key?
You ought to have some good fun with that one and your other set-up, you could string wires out to the shed and have a dinner call communicator! ;c)