Yeah, sadly that's going to get messy, as you're right that the sync box doesn't pass ARC back through. About the only way I can think of to deal with this would be to manually change the connection to the sync box when you're streaming from the Apple TV, as it doesn't need eARC in that mode since it plays directly to your HomePod(s). However, you'd have to take the sync box out of the loop when you plan on using any other device that's connected to the TV, such as your Nintendo Switch.
I suppose in theory if you could find an HDMI splitter that supports eARC, that could work too — use a 1->2 splitter and connect the Hue Sync box to the non-ARC port, and the TV ARC port to the splitter's ARC port. You'd have to test it to find out, and you'd probably still have to switch the TV to the ARC HDMI input when using the Apple TV in order to avoid whatever confusion might ensure from playing something to the TV AND getting an ARC signal back from it at the same time.