Backup with time machine without format
According to the Time Machine documentation it requires a HFS formatted drive to work. 

To get around this problem, on my external FAT32 USB drive, I split it into two partitions. One for Time Machine, in HFS, and the rest for FAT32 to share files. 

In you case, since it already has files, I would find a tool to shrink the existing partition, then create a new HFS one in the now free space and voila! Your problem is solved. If the existing filesystem is NTFS then Windows Vista and above I think will happily shrink it. If FAT32 then have a look here.