Update, I removed Translatepress and tried installing WPML again. This time I'm happy to report it installed without giving me the white screen issue. I was also able to save most options in the application, however I still get the "Error: data not saved" error when trying to save setting changes under WPML > Settings > Post Type Translations and Taxonomy Type Translations.
I won't know any more about if it works or not until I actually translate a full page and do some testing, which I will try and get to this weekend.
If possible, can I ask if any errors were spotted in the logs that the engineers were checking before? They mentioned:
"Looks like the error is related to a global variable "$wp_locale" which has something to do with translation, not sure exactly what it does.
But the logs suggest "$wp_locale" is null at the time of the error. I can't tell if this is exactly related to the error the pub is having but it is fatal."
I will take the information I have, plus your reply back to WPML and see if they have any suggestions, and go from there.
Nice that the white screen issue seems to have cleared up after renaming the uploads directory and setting permissions to 775!