Why Catalinas, not bombers, carried the search burden

Could bombers or another squadron provide the reconnaissance that Hawaii lacked?

Not sustainably; the Catalinas were the operational search aircraft, and there were too few of them and their crews for all-around coverage.

Catalinas carried the long-range reconnaissance burden around Hawaii. The bomber force was not being used to provide comparable continuous ocean coverage. B-17s arriving during the attack were part of an attempt to strengthen the air groups, not proof that Kimmel already possessed an adequate search force. With enough Catalinas and crews for only about sixty degrees, the command had to concentrate effort instead of pretending it could cover all 360 degrees.