Here's a possible answer.
I designed a functionally similar circuit to wake my APC inverter when the
fridge wants to run. Mine requires some logic circuitry because it needs a
hack to handshake with the APC Start button and buzzer to wake it when on
battery power, so the start circuit is more complex than you need. APC
declares that waking on battery power, while possible, is not a "normal"
operation. They also scolded me for finding and using their "proprietary"
A current sensing relay will detect when the fridge shuts off and command
the APC to sleep, or it could return a genny to ECO if combined into Pete's
and Ron's circuit The relay has an LED that indicates when the fridge is
drawing current, which tells me to wait a few minutes when switching its
plug between grid and APC power.