If you still have access to the old site, reduce the size of your backups by breaking them in half. Do half course backup, then other half course backup, or if you have lots of student data, do course first, then do user data as two different backups. Restore the User data into the existing course. Or do sections 0 to 5 and half the user data, then sections 6 to 10 with rest of student data. Or use any configuration that ends up with your backup files a lot smaller. Lot more playing around, but will achieve the end result.
In the meantime, talk to the service providers and see if you can get an increase in size of uploads. Go to 256MB if you can, courses will easily get to that size as they become more sophisticated. Eventually that will need to double again, which is why I am suggesting that where possible, self host on your own dedicated Moodle server.
As for fixing the mbz file manually, the mbz format is simple to change, rename it to zip and unzip it. The important files will be the xml files and where there used to be only one, there are an awful lot more now. Trying to edit, say by removing the images and video if any, the xml file references to the images and video, then rezip and rename back to mbz is a nightmare, miss one image reference, get one step wrong, and the backup is a dead duck. I wouldn't recommend it unless you are really familiar with xml and the Moodle backup structure.