I understand your wanting to keep using the string stops, I use to use them exclusively. I simply urge you to try the bead type plastic stoppers (I forget their proper name but they come in a bunch on wire holders). Their shape and smoothness should prevent hang-ups on the reel. I avoided using them for a long time because I couldn't figure out how they worked. Simply put about 3 or 4 inches of the end of your line through the wire loop and pull the bead off the wire right onto the line. You can then adjust it to wherever you want. It is the fastest stopper to put on that there is, as far as I know.
GET THE NET HAROLD, GET THE NET!!