I ran the into the same thing when we had to bring in "time of day" also.
You'll have to wrap the item with a TO_DATE() in all of your SQL and PL/SQL usages.
schema.my_package.my_procedure( TO_DATE( :X_DATE_MONTH, 'MM/YYYY' ) )
Haven't tested it, but perhaps try adding a computation with TO_DATE to set the column variable to a date first. This should convert it to a date and then allow the built in DML to proceed.