Short answer is no.
From what I'm reading, it is almost impossible to do what you're describing given the environment you're in. There is likely a policy that an administrator at your organization/ school assigned either to your account (school email) or the device itself which disallows access to crosh and/ or other parts of Chrome or select settings on the device itself.
It's been a few years since I have been on a managed device/ account with similar organizational settings you're describing, but your organization (meaning the school and/ or your administrator) ultimately control the access to the device you're on and the network you're likely using at school, which again dictates the things you can and cannot access. Since no one school's settings for their computers or accounts will be alike, it's highly unlikely for someone to be able to recommend a way to circumvent any of the changes or policies you've described; which you probably shouldn't be trying to get around to begin with.
My advice is if you want to have full freedom over the things you want to do, then get your own Chromebook or device to tinker with, along with using your own personal account for it. Use the school computers and your school issued account for what it's intended for: Education.