I had the same issue when replacing the original Hunter module which failed. It had a hot in to the module, light and fan out from the module, and one neutral common to the fan, light, and control module.
The LED lights supplied with the fan were dimmable, 40W equiv, IIRC. When I installed the Inovelli switch and module, the lights were fully dimmable, but that night, once it got dark, I noticed the lights still glowing dimly. I replaced the bulbs with 60W equiv, and they now go completely off.
If I read your description correctly, it sounds like you have non-replaceable bulbs. In that case, a bypass should provide enough extra load to allow the lights to fully extinguish. This is a somewhat common issue with dimmers and low wattage LEDs; and the use of bypasses a common solution.