I have used both the supplied lav mic and Røde GO lav mics with the Zoom F2, but I get no audio using the Sony ECM-322BMP.

The Sony ECM-322BMP has a 3.5mm TRS, so similar to the lav mics, even with a locking connector similar to the supplied mic.

I wonder if it is a pinout issue, so the Tip-Ring-Sleeve is different on the Sony vs Zoom/Røde?