Time machine estimates size of backup much larger than needed

What is the exact wording of that Time Machine dialog please? A screenshot would be preferable.

A message saying "Estimated size of full backup: 5.54 TB"

Time Machine overestimates by design, sometimes too much. That's just the way it is. The screenshot of any dialog will help advance troubleshooting.

You may also need to describe your installation in greater detail. I am confused by the presence of a 1 TB startup drive and another 5 TB "internal drive". If TM is backing up both of them (as it will by default) a 3 TB backup drive is clearly insufficient.

It's also not a good idea to partition a backup drive. That is likely to be unrelated but as I wrote we need a thorough description of your installation, which appears to be more than a simple one.

One obvious standout is the 8TB drive for audio / video work. Unless you exclude it, TM will back it up. That alone is unlikely to be successful with a backup drive of capacity substantially less than 16 TB, an estimate predicated on Apple's recommendations, here: Back up your Mac with Time Machine - Apple Support. As with most such "rules of thumb" it is less than precise, and as it applies to your installation, an overly simplistic one. You can probably get away with a backup drive of less capacity than that, but suffice it to say its present capacity is insufficient.

The estimated size of a full backup shown in the last screenshot reflects the bare minimum of one and only one complete backup. More than that would be comfortable.

Exclude the drives you don't want to back up by clicking the + button in the last screenshot.

Exclude files from a Time Machine backup on Mac - Apple Support explains, but the word "files" in its title really ought to be "items" as the rest of that document correctly states. You can exclude anything — files, folders, entire drives.

Expect TM to remain in the "preparing" stage for the better part of a day. Interrupting it will require it to begin all over again.

If Time Machine remains in the preparing stage on Mac - Apple Support

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Thanks for the update. Subsequent incremental backups will take only a few minutes, as usual. TM needs to spend a long time in its "prepare" stage in circumstances such as you encountered.