Well, there are lots of reasons to doubt her. She’s the voice of the establishment that chose this path. One that we’re learning is far more complicated than publicized.
If she was lying about not having enough ships, Picard would have easily been able to find out the truth. But nothing of the sort happened, so I'm taking Clancy at her word. She was already going to deny Picard his request whether she had an available ship or not, so lying about how many ships Starfleet had would not have mattered.