Janeway talks about it like energy was the problem
The way I see it, recycling the watch would have used energy, not produced it.
My thought is that in the situation the ship was in, it would have been bad form for the First Officer to be giving the Captain anything. Even something replicated months earlier. She ordered him to recycle it for the sake of appearances.
If recycling could have resulted in the creation of power/energy (as mentioned above) Janeway could have order a mass of metal equal to the watch be recycled. The books would have balanced.
She wanted the watch itself gone.