I believe you are understanding this in a wrong sense.

You do not

Need LWW if keys are immutable.

You should

Use keys that are immutable if DB is of type LWW.

The pre-condition here is not that "keys are immutable". Precondition is "DB is LWW"; and conclusion is not "Need LWW". Conclusion is "given pre-condition DB is LWW, you need to make keys immutable"