Tom,
You have stated, I think correctly, that you have two energy inputs: the electrical energy that drives the displacer, and the "heat" in the ambient air. The energy in the air is only available if you have a temperature difference. (I do not believe there is any way around this fact).
In your engine, you are creating a temperature difference (by doing work via the turbine) using the energy that results from that same temperature difference. Does this analysis make sense? I lack the education of others posting here, but I believe this is the crux of the matter.
Dale.