Hmm, it sounds like you’re using the dimmer to actually control the smart bulbs as a “regular” load, not with Z-Wave association, or possibly you have some combination of both. Here is the recommended setup for using Illimun bulbs with Z-Wave association from an Inovelli switch:
Notably, you’ll need local control disabled, which will prevent physical manipulation of the dimmer from affecting the load, which is normally the outcome you want for smart bulbs.
The GE Aux switch is not Z-Wave, which I’m mentioning because it sounds like you’re trying to use Association there. You can configure the LZW30/31-SN to work with an aux switch (see the appropriate Z-Wave parameter for that device), dumb switch (same parameter, different value), or other smart switch (this would also be Association), but the aux/add-on switches don’t have “smarts” of their own, so Association is not an option. There are some differences between all three of these options that is probably spelled out somewhere I can’t find right now…
The LED bar is intended to stay on (or however you have the switch/dimmer set) when local control is disabled. You can manipulate its appearance (but not its apparent “level” representation; just color and brightness, with 0/off being one option) via Z-Wave parameters if you wanted. So, I’d say your interpretation is correct.
Not sure about the ST/HASS integration. Does something in particular break? Remote control would disable Z-Wave on/off and level control, so if that’s what you’re trying to do, you should find the same issue regardless of whether the change originates on ST or HASS (though I’ve never used this integration, so maybe you’re trying to do something else that you think should work).