First drape the toga on a form. Now a toga is normally one long piece of fabric so once you figure out how you want it to look you have to figure out where the costume is going to break in order to come off. If you are doing a one-shoulder style toga you probably only need to cut it on the side seam opposite of the shoulder. (And if you can talk to the director about the blocking so you figure out which way the toga needs to come off stage. If the toga is going out stage left, put the shoulder strap on the left.)
Now that everything is draped you need to tack the heck out of it so that it will stay toga shaped without needing all the tying and pinning that real togas need to stay a toga and not a pile of fabric. It may be helpful to have a flat base that you can tack all the pleating and draping too.
I would first try snap tape on the breakaway side, but you may have to experiment. To create an anchor point for the fishing line, I would get clear suspender buttons and sew them to the toga in strategic places. You'll probably need 2-3 anchor points. For extra sturdiness use 2 suspender buttons per anchor point, one on the outside and one on the inside sewn one together. It's easy for stitching to rip out of fabric, but you'll have a hard time pulling a button through fabric, right? The fishing line can be sewn or tied to the buttons.