How to automatically stagger transaction log shipping to minimize bandwidth usage peaks?
DBCC SQLPERF (LOGSPACE) will return the size of the log file in MB &amp; the current percentage used. From that you can calculate the current utilised portion of the log file in MB.

That by itself will not be sufficient to calculate log backup size, not least because you are using backup compression. However collecting this data over time &amp; correlating it with the size of the backups might reveal a linear relationship.

You could insert a step into the Log Shipping backup job to record the size of the utilized portion of the log file and a another step after the backup to record the new size of the log file and the size of the backup.

Running this for a period and analyzing the results could allow you to estimate the size of the backup file.

Not the most straight forward method and to be honest it might not even work. If you are interested in giving it a go I would be interested in helping.