Hi emmakaur07,
I can see a case for both options. Two thoughts:
1) Which makes your feel lighter? Moving on to the next phase of your life, or getting it down on paper and releasing it first? Writing the letter should feel like a release rather than feel like something that makes you even more upset, and you should also make sure that you don’t do it from a place of anger so that you’re taken seriously and don’t burn any bridges (Yue’s suggestion above is great).
2) Is there a middle ground, where you could share your thoughts with your manager’s manager without it being an official “letter of grievance”? Or, keeping the letter short and sweet rather than long and detailed.
Good luck!