I think your suggestion is a good one. However, I'm not prepared to spend the time and legal fees to take them to court and I don't make empty threats.
I just got off the phone with them. They seemed very professional and they said they typically charge 25-30% of the collected amount, but they usually don't do one-off collections. They typically work with companies that have at least 10-15 non-paying customers.
I understand that you find collection agencies "scary" but at this point I've exhausted all other avenues. My responsibility is to my company, not their company and staff.