So here's one of those things that I need to document, or I'll forget. One of those blogposts that I'll find through a search engine in a year ;)
The solution was actually pretty simple. Create an unmapped volume inside the mapped volume:
- ../project:/var/www/html/project:cached - /var/www/html/project/web/js
Now when I ran Encore, I had no more errors and everything worked as it should.