First things first:
The tool, Barman indeed is a really good tool but for your used case I seldom believe its coming out as a productive one.
Second, you need to redo some of your backup strategy work. Looking at the backups your are consuming, and I do not know what your retention is(i'm assuming it won't be high looking at the size of the backups) here are my 2 cents:
Compressing backups may save time and space but it adds an overhead and time when your want to restore(again this doesn't apply to smaller DB's).
Taking daily backups of a DB which is TB's(might grow as well), is not a good option when you compare it with having incremental and logs on top of it.
Have a base backup every week, differential daily and then logs on top of it. You can always fine tune this along with the retention.
I'm not sure if you environment supports snapshots but this is another way out and really speeds up the restore and backups for you, its snapshots(at machine level, storage level etc.) In essence take the snapshot, tune your pg_starts and pg_stop along with the snapshot timings and then put archival logs(this is if someone archives their logs)
The above would help in speeding up the process and also give you back space.
Now coming back to the tool. Pgbackrest is an excellent option to perform all of the above (except 4.). I see it working for your used case better than barman if you do not want to redesign the backup strategies(which I would recommend doing irrespective of which tool you use)
I would recommend against taking a backup and then zipping it, just to save space, this is taking more time even after the backup is done and restorability takes a hit. Futuristically this approach will not scale well too.