I'd break this out into separate steps. The file requirement may be there, but consider this: as long as every row generates a file, is it truly a deal-breaker if it takes perhaps five or ten minutes to create the file? And is this worse than the knock-on effects you'll see from running a SSIS package from within the trigger?
So:
Step 1 : the trigger simply inserts a row into another table with the information required for the file to be created.
Step 2 : modify your SSIS package to have an extra early step that polls that table for any new entries, creates the files as needed, then marks the entries as completed (or removes them completely, but personally I like audit trails).
Step 3 : add a scheduled job to the server that runs that SSIS package every 5 minutes.
If you don't want to modify your SSIS package, you could instead create a stored procedure that polls the table and executes the package, and schedule that in a job. The main thing is to get away from the idea of firing the package direct from the trigger.
The method above will also minimize the impact of potential problems such as the file destination being unavailable for some reason.