Can the "Declaration of the Causes and Necessity of Taking Up Arms" by the colonists to the king be considered a radical endeavor?
I assume that you are asking about the "Declaration of the Causes and Necessity of Taking Up Arms" and have changed your question accordingly. I would argue that this declaration was not a radical endeavor but was, instead, an attempt to be conciliatory towards England even after the fighting had started.
I say this because the declaration does not declare independence or push for any truly new ideas about government. It does not denounce the king or the idea of monarchy. Instead, it blames the problems that the colonies are facing on the ministries and on the Parliament. It declares that the colonists have no desire to break away from the British empire and that all they want is the same rights that Englishmen have and expect.
This is not a radical document. It is one that speaks out strongly against specific actions and policies of the British government, but it does not advocate anything radical like independence or defiance of the king.