I think that my school (UCI) has the same letter packet service as you guys and I didn't include the cover letters with my recs and I just got an interview invitation from Pitt. I had the same worries as you in the beginning, but after contacting some schools I found that they don't mind you using your schools cover letter as long as it has your info on it. One school's cover letter (I think it was Wash U) asked specific questions such as: "rank this student by percent, compared to other students you have worked with" and they still said it was okay if I don't include it.
Don't take my word for it though, just contact the school and make sure...good luck!!