I can't trash the Mac OS BIG it says Can't be moved to Trash because it's opened.

Where is the application called " Install Big Sur " located on the computer ?

What was the user doing at the time it was attempted to move to the Trash ?

Force Shutdown of the computer. Even use the Power Button to shutdown the computer ( Hold Power Button until it goes Off )

1 - Restart in Safe Mode. This will perform a Disk Repair, clear cache files and only load Apple Software, extensions and fonts. The boot up will be slow and can take some time - Normal.

2 - Safe Mode will also eliminate Third Party Software, extensions and drivers from loading 

3 - Does the issue present in this mode ?

Sorry but this situation may require a trip to Apple Repair and Repair Status Check - Official Apple Support

It appears the ungraded process is already in progress but can not move forwards for the lack of empty space on the drive which you already mentioned

Thus has basically bricked the Maxine for now

Welcome and you did all the advanced work to to fix this issue 👍

Onto the original issue that was the cause which is Empty Space

From memory the computer may require about 35 GB empty space from High Sierra and higher and 44 GB empty space for Sierra and lower

I suggest a minimum of nothing less than 50 GB Empty Space.

Rebuild the Spotlight index on your Mac

Or Force a Reindex via Terminal Command “ sudo tmutil -E / “ without Quotation Marks 

What is “Other” storage on a Mac, and how can I clean it out?

Free up storage space on your Mac

OmniDiskSweeper Safe to use

GrandPerspective 

How to delete Time Machine snapshots on your Mac

See used and available storage space on your Mac

Locate backups of your iPhone, iPad, and iPod touch

Notation - If the user is using a cloning software like Carbon Copy Cloner - suggest tweaking the Safety Net Feature in this software. It may be making additional Snap Shots that are not being Cloned to the Eternal Drive. If this should be the case, these Snap Shot could be using additional space on the drive 

The final word from Apple on Managing the " Other/ System Data “ Category

Other / System Data: Contains files that don’t fall into the categories listed here. This category primarily includes files and data used by the system, such as log files, caches, VM files, and other runtime system resources. Also included are temporary files, fonts, app support files, and plug-ins. You can't manage the contents of this category. The contents are managed by macOS, and the category varies in size depending on the current state of your Mac.

Shutdown the computer using the Power Button until it goes off.

Pause for 2 minutes then restart normally

Try to move to Trash and empty Trash