It's going to be hard to verify every detail about a personal interaction on set, but we can say Bermann's version doesn't match what is verifiable. The shows were shot out of order from how they were aired. Crosby's chronologically shot final scene was in Symbiosis, where she's seen waving goodbye in the cargo bay.
Remembering the wrong episode would be one thing, although even that dampens his tone of a "cherished memory."
He completely invented that she gave it to him, instead of him taking it from her. That also drastically shifts the tone.
The story changes from "a dear friend gave me a gift on her onscreen death that I cherished for decades" to "I took a one of a kind memento from someone on their last day at work on something iconic and kept it for myself. I stared at it for years."