score:9
Taking the last of the question's points first - I think we can free the Bermudian shipbuilders and the Bermuda cedar from particular blame because the follow-on Cuckoo class ships suffered much the same way and they were all built in British shipyards with European woods. If the materials or construction process were purely at fault, you would perhaps expect to see more of a difference.
The Royal Navy lost far more vessels to the 'dangers of the sea' (wrecks/foundering/accidental fires) than they did to enemy action. This was largely a result of their policy of keeping ships at sea, year-round, in all weathers in order to maintain what A.T. Mahan later called "Sea Power" – keeping control of the seas.
During the French Revolutionary and Napoleonic wars, the RN permanently lost just 10 ships to enemy action (a few others were captured but subsequently re-taken). In the same period they lost a total of 344 ships to the sea; 254 were wrecked, 75 had foundered and 15 had burned/blown up. As a British sailor, you were almost safer in a battle than you were sailing to and from one.
With that in mind, were the Ballahoo and Cuckoo-class schooner losses particularly notable?
If we look at a couple of slightly larger schooner classes used by the RN in the same time period, the pattern of losses there isn't all that different.
Taken in that light, the losses of the Ballahoo and Cuckoo-classes might not be all that exceptional after all.
I think there are two key elements that affected these vessels (which might well apply to the larger schooners too):
[Describing the schooner Haddock] An attempt to harness the expertise of Bermudan builders who were renowned for their fast-sailing small craft, this class was primarily designed for dispatch duties rather than as cruising warships. Lightly armed, but forced by their role to go in harm's way, they suffered heavily from both the dangers of the sea and the violence of the enemy.
British Warships in the Age of Sail, pg.358
References:
British Warships in the Age of Sail, 1793-1817, Design, Construction, Careers and Fates, R. Winfield (Seaforth, 2005)
A Social History of the Navy 1793-1815, M. Lewis (Allen & Unwin 1960)