An immutable object is initialized by its constuctor only, while a singleton is instantiated by a static method. How to make an immutable singleton in Java?
You're being unnecessary complicated. To be immutable an object must be unmodifiable once it is created. That's normally interpreted to mean "modifiable only in the constructor", but if you were to create it another way that would still make it immutable. As long as your object cannot be modified after it is initialized then it is immutable. You can consider setting up the Singleton instance to be part of the initialization.
Most of the benefits of immutability are irrelevant in Singletons.