Issues with Time Machine backup on external drive

you're going to have to clear some space on the mac's drive. not only can you not update macOS, your Time Machine will not be able to backup the Mac. to make the TM backup, there needs to be enough free space to make a "local snapshot" which needs to be placed on your internal drive BEFORE it gets sent to the external TM drive. and, quite frankly, with only 2.55GB of space on the Mac, I'm very surprised that the Mac has continued to function at all. it's recommended to keep roughly 20% of the drive's space to remain as free space for your Mac to work properly.

I would recommend that you purchase an external HDD or SSD of sufficient capacity and move your larger files (videos, pictures, documents, etc...) to it. once they are on the EHD, you can simply erase the same files from your internal drive. and, if it's just going to be used for storage, you can use an HDD and save a few bucks over the cost of an SSD. FWIW, i personally keep very little on my internal drive.

also, you may want to read these two articles:

Optimize storage space on your Mac

How to free up storage space on your Mac

To add to the good advise from @jefferythefrog

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

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.

Notation - it is suggested to keep at least 15%  to 20% of the Drive Total Capacity as Empty Space for good operations of the computer and for additional updates to Monterey to download, expand and install.