Thanks all! I'll be using one (or more) of these suggestions.
I had already tried drilling the batten and sewing it down with waxed thread, and that did work for a year or so. However, here in North Carolina I rarely take the sails off the boat, as we sail year-round (Glory! We moved here from Rochester, NY.), and I wasn't watching it. We're on a lake with twitchy, gusty winds, and I guess that flogging the main sawed through the thread.
The use of sail repair tape across the strap at the leach sounds like an interesting approach. Thanks again.