The "antique" frames that I have seen/used have a strip of heavy fabric (pillow ticking) nailed to the long poles. There are loose strips hanging from the poles at your right and left and those will help keep the quilt taut on the crosswise. Your description sounds like you will need the strips to fasten to the quilt on the sides and then around the horses.
If you have the ticking strips on the long poles, you can use pins rather than tacks to attach your quilt sandwich to those ticking strips. You have the sandwich construction correct.