Actually, now that I think about this some more, I think you’ll still encounter other problems.
Even if you do end up resolving this, I think you’re still going to encounter other CSRF and CORS-related issues.
I don’t think this idea is going to work as is.
I’m starting to come to the conclusion that if you’re running both projects on the same system, you may end up needing to run them both behind an nginx instance that allows the browser to be issuing both requests to the same server. You can then configure nginx to separate the traffic between the two projects. (And, it would let you separate this traffic based on name or url, and not just port.)