The polarity indicator is connected between neutral and ground. When it lights up it indicates that there is voltage between neutral and ground.
First, the secondary winding ground needs to be connected to your secondary neutral at the isolation transformer. Thus, there can be no voltage between secondary neutral and ground. Check that.
Second, make sure your grounds are correct.
The isolation transformer has two grounds and they should not be connected. The ground shielding the primary windings gets connected to the incoming power line and nothing else. The secondary winding ground becomes your new boat ground. The two grounds should not be connected onboard. This would defeat the whole purpose of an isolation transformer. Non-marine isolation transformers tend to connect the primary and secondary grounds. Don't use those.
If your grounds are correct, the indicator light could be lit by some induced stray voltage. It does not take much current to light a LED. Use your meter in low-Z mode to eliminate inductive voltages and reroute your wiring.
Your inverter will do its own ground/neutral switching. When on shore power, the inverter should leave neutral/ground disconnected (since they are already connected at the secondary windings). When inverting, the inverter should connect downstream ground and neutral since it becomes the source of power.
