can I back up my iCloud on a different device if the device I need to back it up on won't work

You can't backup on the device, since the screen is not functional. And while you could try to backup to a computer, by using iTunes, I'm pretty sure that won't work as you'll still need to enter the passcode on the phone to backup.

But everything you already have in iCloud will stay in iCloud and can be restored to either a new device if you buy one, or if you decide to get Apple to replace the screen on your iPhone (assuming the only issue is a bad screen), you'll be able to restore what you have in iCloud to your fixed iPhone.

You don't back up iCloud. You back up an iOS device like an iPhone.....to....iCloud.

Are you saying that you need to back up your iPhone 12 Pro.....but the iPhone 12 Pro is not working? What exactly is not working on the iPhone 12 Pro?

No. If the device is damaged or inoperable, it cannot transmit data anywhere and certainly not using a different device. The time to ensure no loss of data is before something bad happens.