How do I backup a site when it weights is more than 200MB and host can't do it?
Refer to my answer to "Which tables are safe to clear?", which should help to get the size of your database reduced.

And/or use the Backup and Migrate module, which also support exclusion of selected tables (typically the ones that are safe to clear also). Here is a quote from its project page (bold markup added here):


  Back up and restore your Drupal MySQL database, code, and files or migrate a site between environments. Backup and Migrate supports gzip, bzip and zip compression as well as automatic scheduled backups.
  
  With Backup and Migrate you can dump some or all of your database tables to a file download or save to a file on the server or offsite, and to restore from an uploaded or previously saved database dump. You can choose which tables and what data to backup and cache data is excluded by default.