Can I manually reset the "internal status flag for a full database backup"?
Your primary goal here is to ensure your ability to restore the database.  As long as you have the complete chain of your log backups from the last full to now, you can perform a point in time recovery and the error is merely an inconvenience.  Differential backups are not required for full point in time recovery, just provide you with a shortcut for larger time frames.  However, in the absence of your differentials, you'll need all the log backups to cover the gap.

Otherwise, the only way to properly reset this is to execute another actual full backup.  I would recommend executing this as soon as possible which, depending on the size of your database, should be something that could be accomplished overnight in place of your normal differential.

Also, I would strongly recommend that you apply Service Pack 4 for SQL 2005 to make sure you have the most current hotfixes/updates for your platform.