Do you mean that the application is storing a files as blobs in the MySQL database, and/or creating lots of temporary tables? Or that you just want temporary files - themselves unrelated to a database - to be stored in MySQL as a backup?
I'm not sure that trying to use MySQL as an net-new intermediary for backups of files is a good idea. If the app already uses it, thats one thing, if not, MySQL isn't the right tool here.
Anyway. If you are interested in capturing a filesystem at point-in-time, the answer is to utilize LVM snapshots. You would likely have to rebuild your server to get your filesystems onto LVM, and have enough free storage there for as many snapshots as you think you'd need.
I would recommend having a new mount point just for this apps temporary files. If your MySQL tables are using InnoDB, a simple script to run mysqldump --single-transaction in the background, and then the lvm snapshot process, you could get these synced up to less then a second.