isAccepting is not a very clear concept, I also don't like that much that it is used as signal to re-fetch the data.
It basically adds a functional dependency where there shouldn't be one.

I think it would be cleaner if accepting the edit would transfer ownership of the command-reference from the inline completion controller to the async accept method (which is then disposed when the accept method finishes).
This could be implemented using ref-counting. Maybe there is even a utility RefCounted for that:

class RefCounted<T : IDisposable> : IDisposable {
  /** Takes over ownership of `value` */
  constructor(public readonly value: T) {}

  /** Creates a new reference to the data. */
  clone(): RefCounted<T> { ... }

  /** Disposes this reference. If it is the last reference, disposes the value. */
  dispose(): void {}
}