It sounds like you didn’t buy into Bluesound’s wireless speakers yet or you would simply place the NODE by the ATV4K & projector and configure it to play to the pulse soundbar, etc. under the screen… So here’re a couple of suggestions/hacks.
For a cheap/easy fix, you might consider and be surprised by the sound produced by a pair of HomePod minis… at $99 each, the pair will cost less than half of what you paid for the ‘NODE. I happen to use an Atmos-capable HDMI-connected soundbar for watching TV, but I have a pair of HomePod minis in the adjacent room… when I tried them out as a “soundbar alternative”, using the default audio output feature, I was pretty impressed! So was this home theater ‘blogger guy.
Before you spend any more cash though (here comes the hack), try creating a HomeKit scene to mitigate the hassle of manually connecting the NODE via AirPlay each time you’re watching the Apple TV. I believe you could add the NODE and Apple TV 4K to your HomeKit setup as accessories, then try creating a scene, call it “Home theater”, that includes both. You can play with supplemental settings details in the scene. Henceforth, a “Hey Siri - Home theater” should wake up the ATV4K and link the NODE via AirPlay.
I don’t have any stand-alone AirPlay speaker accessories in my HomeKit setup to test what I just suggested, but I often want to have the HomePods playing a Pandora station while also hearing it through the ATV4K-connected soundbar… This was a two-Hey Siri command sequence: “Play station”, then “Also play this on Apple-TV”. I created a “Music everywhere” Scene that does join the HomePods and ATV4K via AirPlay and resumes what was last playing on the HomePods. If I used an Apple Music subscription (instead of the Pandora stations I’ve tuned over the last 12 years) I could also specify a particular music selection, but that’s me running into the non-Apple product penalty akin to your default NODE challenge :-).
I did make a product request of Apple to let HomeKit scenes execute typed Siri commands, but that hasn’t happened so far.