Burton batted with proficiency and patience to score 52, the sole half-century of the match, while Pepperell was a menace for Kamo, ending with figures of 4-41.
Kamo made their way to 197 all out, a total that kept both sides in the contest.
Unfortunately for WBHS, they couldn't quite overcome the more senior Kamo side.
Matt Kingi (46) and Atul Nautiyil (47) provided the backbone of WBHS' innings, but the bowling of Kamo and Singh (3-50) kept wickets coming at regular intervals.
The high school side ended on 190-9, just seven runs shy of a Greg Guy title.
In the other Greg Guy Shield match, Mid Western Reserves had a comfortable 48-run victory over City Reserves.
Mid Western set City a target of 236 on the back of 153 runs from their middle order.
City struggled from the outset, losing wickets consistently. Only wicket-keeper Vaughan Witten stuck around, scoring 54 before retiring hurt.
The reserve grade sides now turn their eye towards the opening round of the T20 competition at Kensington Park.
In Dargaville Shield action, Mid Western held their nerve for a tense five-run win over City.
Electing to bat, Mid Western got off to a pedestrian start with only 32 runs coming in the first 11 overs.
Both openers were gone after 16 overs but that brought Northland pair Kurtley Watson (66) and Tyler Lortan (65) to the crease.
The experienced pair paced their 106-run partnership superbly, but both fell within four overs of each other which put the skids on Mid Western's innings.