Cannot partition external drive
The error that your getting...
Exec format error

Is a response from the SATA controller (USB to SATA bridge) to diskutil that there was an error when you tried to format your drive.  This points to a failing drive.
fsck is a utility to fix A corrupted file system and will not have sny effect here.  Looking at it from a simplified perspective, in order for fsck to work, you have to have an existing filesystem in place for it to fix.  formatting a disk is the act of wiping away whatever was there in order to install a new filesystem.  It’s like bulldozing your property to build a house. In this case, the guy doing it reported a problem that prevents you from doing it (they found a fault line).  Running fsck at this point would be like calling the plumber to come fix the pipes because you can’t clear away your property.

External Hard Drive no longer working

Basically, you need to replace that drive.
From your further attempts:

Then I had no issues creating multiple partitions under MBR, but any attempt to go back to GPT led to the message above re backup table and failure to create multiple partitions under GPT.

This further lends credibility to the diagnosis that your drive is failing. Where the tables are stored is in different places on the drive; the “whole” drive doesn’t have to be completely inaccessible for the drive to be considered bad.
I’ve seen this many times: folks will spend countless hours trying to get a drive to format that keeps generating errors. Then, they’re successful once, get it formatted, copy data over, do work, then suffer a catastrophic failure because the drive was dead (dying) to begin with.
Formatting a drive is a super easy task. If it takes you multiple utilities and days of effort, something is very wrong.  Don’t risk your data; get a new drive.