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.)