Fiber transceiver operation
I set up the experiment again and got some positive result so I thought I would answer my own question.

Using both channels on my DG1032Z waveform generator I create a differential square wave signal that is fed into the transmitter side of the fiber module. The receiver side is connected to an oscilloscope.

The waveforms below do not look very clean but that's because I just soldered on some wires to the module. It's probably possible to get cleaner results by using a proper connector for the SFP module and do all the termination correct, but this was just a test to see what actually happens.

Here is the output waveform from a 10MHz input:


1MHz input:


200kHz input: