I don't think there is an easy way to achieve your goal. What you can do is to use this script
to get list of tapes written during the latest session and, then, check free space left of them. Thanks.
If there was only one job per tape it would be possible to determine the job backup size by simply subtracting free space from the tape media size. But there could be multiple. Therefore just the tape free space is not enough to work that out. Or am I missing something here?
I can easily find this information by going to Veeam Console and clicking on the job. If it's kept somewhere I assume it should be possible to query it. Or is it not the case?
Determining the size of a backup seems like a very basic functionality and expected to be supported regardless of whether it's a backup, tape, copy or a replication job. I'm surprised it's not there. Maybe worth opening a feature request then. What do you think?