No experience with double-skips here, but my DD's school used the nebulous fear of gaps as an argument against a single skip. Once we'd gone around that issue for a few laps, it became apparent that it was basically a fear of an unknown, at which time it became apparent that the only rational response was "get the data!"
My recommendation would be to have him take a normed achievement test at end-of-second-grade level to identify these gaps, then begin teaching the areas of weakness. You can't address what you don't know.
We'd expect that he'd naturally pick up anything he's missing as the class progresses. There are potential consequences while that process unfolds, though. He might ask questions that seem obvious to others, which can be used as evidence to himself and to others (peers, teacher, school admins) that he doesn't belong.
