That's a huge amount of data but there might be a possibility for you to FXP from one server to another via FTP.
As I understand it FXP allows you to connect to both hosts, and transfer from one host to another in an interface that is drag and drop. (Doubt permissions will be pulled with it)
When I tried to do this it didnt quite work that well.
The only way I can think that you can do this reasonably is to have the datacenter back up the data to another drive, and have that drive shipped to the other datacenter.
*requesting them to keep a backup of the data until the site is functioning again.
That really is a large amount of data to tar, then wget and as you have mentioned, it might be impossible to do from your location due to space limitations on the host's side.
Have you asked them directly?
Somebody with more server experience might know how to backup the permission settings, then move the data one file at a time, then restore the permissions.
But personally it's way outa my league.