Should I write it all in the additional part of the Common App?
Nope. Start editing. Hard.
Or pick a new essay that will fit in the allotted space.
Admissions is only going to spend 8-10 minutes reading your entire application. It isn’t going to go well for you if they stop reading because you took too long to make your point or tried to circumvent the rules.
What are you planning to tell them? I agree that whatever it is should fit in the word count they give you.