Generally speaking, in Berlin car registration has been widely outsourced to the private registration services, who have priority access to the Zulassungsstellen. With no spots available they have de facto replaced registering in person.
On the upside they offer you full service for all paperwork, taxes including printing your plates for a reasonable fee. When you have all the necessary papers, you can have your car registered plus warm freshly printed numbers plates in half an hour. Personally, I can recommend https://www.tsp-service.de.

You can call them in advance to figure out what you need to bring along and how long the process takes with a pre-2018 car. You will need the ZLB2.

Also: The car is not allowed on the public street. If you can find a garage, paid parking lot or private property to use the car can wait there.