You have two pieces of bad shell going on here:

The first one is because the redirection is happening on your machine. The second is because everything after the > is a filename, but you have included random arguments to curl in them.

To solve the first one, package the whole command into a shell literal:

kubectl exec $WPPOD -- sh -c "curl --request GET --header 'PRIVATE-TOKEN: *******' 'https://gitlab.com/api/v4/projects/*****/repository/files/infrastructure%2Fwordpress%2Fdeploy%2Fall-in-one-wp-migration-unlimited-extension%2Ezip/raw?ref=Add_WP_MySQL' > /var/www/html/wp-content/ai1wm-backups"

I would even go so far as to say "don't use redirection," since if you inform curl of the output file, and add --fail to it, then it will avoid writing to that file on server error, which isn't true when using a shell redirection: the shell will create that file, no matter what, possibly making it empty; thus:

kubectl exec $WPPOD -- curl --fail --output /var/www/html/wp-content/ai1wm-backups --request GET --header 'PRIVATE-TOKEN: *******' 'https://gitlab.com/api/v4/projects/*****/repository/files/infrastructure%2Fwordpress%2Fdeploy%2Fall-in-one-wp-migration-unlimited-extension%2Ezip/raw?ref=Add_WP_MySQL'

For the second problem, it's a simple matter of re-arranging the arguments to be compliant with shell syntax:

curl  -O -J -L --request GET --header 'PRIVATE-TOKEN: Z7-RByYpUJcnWU_STpuz' 'https://gitlab.com/api/v4/projects/14628452/repository/files/infrastructure%2Fwordpress%2Fdeploy%2Fall-in-one-wp-migration-unlimited-extension%2Ezip/raw?ref=Add_WP_MySQL' > /var/www/html/wp-content/ai1wm-backups/all-in-one-wp-migration-unlimited-extension.zip

Although in that case, you have conflicting curl behaviors: the -O option is going to write out the file in the current directory, so your shell redirect is only going to receive any messages written by curl, and not the content of that URL

All of this has nothing to do with kubernetes, or a directory, or a copy, and those tags should be removed.