Time Machine could not back up the disk “Macintosh HD — Data” because it is nearly full.

The third screenshot you posted is telling you it cannot partition the container

as one or more of its Volumes is not mounted.

Now look at your first screenshot and look at your external drive in the

left hand panel you can see the Time Machine Backups Volume is not

mounted.

Highlight Time Machine Backups and click on the Mount button at the top of Disk Utility.

With Time Machine Backups highlighted click on Partition now see if you can create the

partitions you want.

Under Internal your Macintosh HD is not mounted you will need to highlight that

and press the Mount button.

What was the exact error message. It may be that the source is too full, not the destination.

And, Mounting Macintosh HD is a fruitless endeavor. It is not meant to be mounted, and not being mounted has absolutely nothing to do with the problem. In Big Sur, Macintosh HD is never backed up, only the Data Volume.

Open Terminal and enter this text,

diskutil list

press Return

this will list all drives for your mac and attached drives and their disk identifier.

note the disk identifier for Macintosh HD and then enter this text

diskutil mount disk?s?

press Return.

Do not know why your Apple Sparse bundle will not allow you to change the partition sizes of

your Time Machine Backup Volumes.

The internal Macintosh HD is not mounted you can clearly see Macintosh HD in the left hand panel is in faint lettering.

What if you highlight Apple sparse bundle disk image just under Disk Images and then click on Partition

are you able to change the partition sizes now.

You need space on Macintosh HD - Data in order to complete the backup. You don't really see the Data volume, but all of your data is stored on that Volume. Macintosh HD is purely the OS and its bundled applications.

Time Machine under Big Sur and APFS makes a snapshot of the volume being backed up, so needs a very small amount of space. For an SSD to function normally, it needs a lot more free space than you have.

You need to move some of your data off of the drive (or delete it if unneeded). If it is an SSD, it will take about a day to reset the storage to be free, again. It may not take that long, but don't expect to see any available space immediately after deleting. SSD's don't work that way.