Your project is ready for deployment. That should be good enough for anyone who wants to check whether you can code at all, you can code cleanly and whether you coding style meshes with your prospective employer's people.
The definition of the word "complete" very much depends what the client defines as a deliverable.the client could define "gimme the code" as a deliverable, or the client states that you haven't a deliverable until the app is running fully deployed and with content provided on some Internet-facing host. Check the scope of work.
I suggest that you avoid and don't use the word "complete" because it's close to meaningless - it is obvious that the meaning of "complete" has a different meaning in the recruiter's mind than in either your mind's or that of the client's. I'll note that the recruiter doesn't get to define "complete" because the recruiter doesn't have a say about the scope of work.
Instead, say something like "the code is written, tested and ready for delivery to client" or "the code is written, tested, deployed on a host, further tested. The app is content provisioned and is about to go live on an Internet facing host. At which point, the client takes officially takes delivery of the work " depending on the scope of work.
Again, anyone who wants to review your ability to code has enough data at this point to draw their own conclusions. You are being hired on your ability to deliver well-written, clean code and if you can provide tangible proof that you can write that kind of code, that proof should be good enough.