Well, it’s not that it’s the best way to approach your use case, it’s one of the only way to get to your desired result 
: Either cross-doc or webhook …
You might want to check this blog post written by @Christiaan_Huizer for the webhook alternative :
It goes with his free webhook pack
Those would be at the moment the only ways to get everything in a 3rd doc AFAIK …
Thinking about though, if you don’t need to interact directly with your Client’s docs, you might also want to check the quite new full page embed feature.
This would allow you to create a “window” opening on your client’s doc (I’m saying a “window” as a full page embed of a Coda page from another Doc is “read only” at the moment… you can find a little bit more info about the future of this feature 
here, as it might interest you regarding the question you asked not so long ago when it comes to single-page sharing )