When FileMaker Server backs up a file, it suspends or pauses the file. User data and requests are not lost - just suspended until the file finishes the backup file.
That is why your users are getting the dreaded coffee cup. This cannot be avoided using automated backups.
The only way I can see to, at least, mitigate the slowdown would be to run a script in the file(s) that would just export just the data into a FMPRO file. You would still have to save copies of the original files to retain layouts, calculations, and scripts in the files any time changes were made.