A good start would be reducing the impact of introduced diseases in the 1810-1845 period, so the Kamehameha Dynasty navy & army remained in the 10's-of-1,000's strong range.
After French attacks in the mid-1800's a treaties with the United Kingdom (despite the Crimean War & Hawaii's stated neutrality) & the United States as a protectorate would be a good 2nd step. However, American pressure for exclusive trading rights could lead to the US-Hawaiian treaty being repealed. Whilst an existing treaty with the UK & a large military would mean that the Paulet Affair would not have happened - thereby not alienating the British.
Assuming the French & British recognition of Hawaii as an independent state still occurs in 1843/1844 (ending Hawaii's "protectorate" status with the British), the US is further alienated by its failure to recognise the nation's independence for a further 6 years. Furthermore, the British sign a treaty of alliance with Hawaii.
With the examples of the chaos of French Monarchic & Republican politics versus the stability of the British Constitutional Monarchy during this period, Hawaiians may have pushed for a constitutional system similar to the British.
Thus, in 1850/1860, the Hawaiians peacefully adopt a Constitutional Monarchy along British lines but modified to suit local requirements.
That's how I'd start it... You can take it from there, if you want.
Cheers!
Guy.
