He's making a metaphor to express the unpopularity of the Bayonet Constitution among native Hawaiians. The metaphor is of something that gets stuck in the throat and is difficult or unpleasant to swallow, he is saying that the new constitution, the Bayonet Constitution, was not accepted and was distasteful to native Hawaiians.
The Bayonet Constitution, imposed by King David Kalakaua, was a new constitution that greatly limited the power of the monarch and enhanced the power of American and European business interests in Hawaii. This was seen as a betrayal by many Native Hawaiians, as they felt that their sovereignty and cultural autonomy were being taken away.
As a result, the Bayonet Constitution was deeply unpopular among the native population and is considered by many to be a low point in Hawaiian history.