All exactly right.
If your lights are on a Hue bridge, that’s a separate network. You can run Hue and SkyConnect side by side, but there’s no crossover.
If you want to keep the Hue bridge you could try a few more routing devices on the Skyconnect. If you don’t need more lights etc. you can get “repeater” devices that don’t do anything else.
There are several recommended products here (towards the end of the post), along with lots of other good advice.