In answer to question 1 (your initial) ... bottom line, no (or I should say to the best of my knowledge). However, you should be able to set the autobackups to 'manual' - which means one could create a bash script telling PHP to run moodlecode/admin/cli/automated_backups.php. Run it anytime you desire via command line.
If it's space that's the issue, what about a mount point (rather than WebDAV) pointed to whatever platform one has. That mount point could be aliased (ie, symlinked) from moodledata/repository/mp3s and if one is running a site backup which includes the data directory, one could exclude the moodledata/repository/mp3s via tar.
Yes, there is a new power that be with IT ... that of the virtual server administrator (the one who has the power to allocate cores/space/memory, etc..). Of course, one could always go along with less than what is required and then tell IT folks (the virtual server administrator) that he/she will have to find a way to increase space in the guest operating system (there are somethings that the Virtual OS still won't do). Cores are easy to do, so is memory ... but space ... that takes work and some expertise (platform depending)! ;)
'spirit of sharing', Ken