If the full backup is too large to get over the network, send that overnight in a package on tape or USB disk, and then send the transaction log backups over the network and roll them forward after you restore the full backup.
It sounds like you have a fairly small amount of data to move. If you can move an 8 GB backup in 2 hours, you should be able to move them all in less than 24 hours and then roll the transaction logs forward after that.
I would setup ROBOCOPY in SQL Server jobs to mirror the full backup directories, and separate jobs to mirror the transaction log directories to the new location. Be sure to use the /Z option for re-startable copies and the /R option to automatically retry failed copies.