This question came up between me and my sister. She was head of the English department at an exclusive private school until her recent retirement. I am a career telecommunications expert.
I think you are asking two questions here. Can the word posthumous be used to refer to an event rather than a dead person? How to construct the sentence correctly using the word posthumous or posthumously with reference to an event after the king's death.
Your use of posthumous weakening is correct to refer to a noun "the government" that happened after the king's death. English 101 an adjective is a word that modifies another person or thing in a sentence. As an adjective posthumous is always used before a noun. Something that happened, was done, or published after someone's death.
Whereas the adverb posthumously must refer back to the action or verb. "Things in the government happened posthumously."
The third possibility is that posthumous can refer to an event rather than a dead person. This third method is only vernacular. In most modern dictionaries you will find the word posthumous always referring to a dead person. However if you search http://www.urbandictionary.com/define.php?term=posthumous in the third definition the word can be used "With hindsight occurring after a particular event. Nothing having to do with death." In telecommunications with regard to an outage that has been fixed we use this term in a phrase such as, "To prevent a future outage additional actions were taken posthumously.
Needless to say my English teacher sister took great umbrage with the third definition when I used it in a sentence to her. With any specialty field of endeavor, especially telecommunications, private definitions of words become common and mistakenly using them with the wrong audience can be catastrophic.
In literature use the traditional dictionary definition of posthumous or posthumously as always referring to a dead person.