Log shipping large DBs - what about the log?
If I understand you correctly, your major problem is the log files during the several hours of backup. From your opening statement I understand that the 1.5 TB database normally runs in SIMPLE recovery, and thus no log backups to do.

Disclaimer: I have never done log shipping on this scale.

Of course, you should ask whether you can get more space allocated for your log files.  If you can, then great.

However, I think a small modification of your plan, providing that you already run in SIMPLE recovery model and/or the risk of SIMPLE recovery model for a few hours is worth it, would ease some of your worries.


Keep (or set) DB in SIMPLE recovery model.
Take FULL backup (5-6 hours) on primary
Restore FULL backup to secondary (leaving in NORECOVERY) 
Set DB in FULL recovery model
Take DIFF backup on primary 
Restore DIFF backup to secondary (still in NORECOVERY) 
Initialize log shipping using 'The database is already initialized'


The apparent advantages are:


No log files to backup during the large FULL backup.
Switching to FULL before beginning the DIFF backup will give you the needed log to start with and its longest growth is probably during the DIFF backup.


Regarding when a log backup can start: 

https://technet.microsoft.com/en-US/library/ms190729(v=SQL.105).aspx 

This says: "A new log chain starts either with the first full database backup following database creation or after switching from the simple recovery model to the full or bulk-logged recovery model."

So, I still believe that this will work as outlined. (Not identical, but I have used a differential backup to cover a gap when log files were lost, so as to establish a new origin for the log backups.)  

(Remember my disclaimer, of course.)