With the outline and the smr_centroids in the same map, you can edit the smr_centroids spatial network. (If they aren't in the same map, add the centroids to the map with the outline in it, and then choose Data>Spatial Network from the data menu).
Choose to select links by intersection with a spatial feature using the spatial network toolbar.
Choose to intersect the spatial network with the "outline". Click "Intersect" to proceed. This selects all links that intersect the edges of the islands.
Turn off all selected links.
Inspect the network by zooming in and panning around. The active links are in green, the inactive ones in grey. Selected links are orange.
You may wish to reinstate a few links that isolate areas completely (say where there is a land connection between two ZIP codes but not where the link is, such as at the north fork of Long Island, illustrated below).
Choose stop to finish network editing. Save your edits.