The technique in this post worked for me
1) Click the "Export" tab for the database
2) Click the "Custom" radio button
3) Go the section titled "Format-specific options" and change the dropdown for "Database system or older MySQL server to maximize output compatibility with:" from NONE to MYSQL40.
4) Scroll to the bottom and click "GO".
I'm not certain if doing this causes any data loss, however in the one time I've tried it I did not notice any. Neither did anyone who responded in the forums linked to above.
Edit 8/12/16 - I believe exporting a database in this way causes me to lose data saved in Black Studio TinyMCE Visual Editor widgets, though I haven't ran multiple tests to confirm.