How do I stop a beadwoven bow pendant from curling at the edges?

You diagnosed it yourself, honestly. Tight tension is almost always the reason a flat woven piece curls, especally on shapes with long thin sections like ribbon loops. Every stitch pulls the beads slightly toward the thread path, and when the whole piece is under that constant squeeze, it domes and rolls. The reason it looks perfect on the mat is that the mat is doing what your chest should be doing: pressing it flat and hiding the problem. The edging row you added helps because it spreads the stress over more beads, but it can't fight the underlying pull, which is why the tips still roll.

Been chasing curling ribbon loops since the 80s, when all we had was Nymo and a prayer. Two things worked for me back then and still do. First, after I finish a piece like that I pin it flat on a cork board with a light cloth over it and let it sit a few days. The thread relaxes and a lot of the roll just goes away on its own. Second, if it still fights me, I loosen up by running the piece through my fingers and gently easing each row before the final weave-in. On the stiffener question, skip the heavy stuff. A dab of diluted white glue worked into the back with a toothpick dries mostly invisible if you keep it off the bead fronts. Test on a scrap swatch first, always.

One thing I'd check before ripping anything apart is wich direction the curl runs relative to your thread path. On my peyote pieces the roll almost always follows the last pass, so if your edges fold toward the back, a final row woven with deliberately slack thread on the concave side can counteract it. I've also found the backing idea is less work than it sounds if you do it in brick stitch rather than a full second layer, and it stiffens the ribbons nicely. As for stiffener, any kind I've tried left a film on the crystal facets, so I stopped using it on anything with sparkle.

I'm not buying that tight tension is 'almost always' the culprit. I've been weaving tight for years because I hate floppy work too, and my flat pieces stay flat. What actually matters is whether the pull is even on both sides. A bow shape like yours has long runs of beads going one direction with nothing balancing them, so even normal tension will bow it. Before you rip anything, try this: hold the piece up and see if the curl follows the long axis of the ribbons or runs crosswise. If it follows the ribbons, tension isn't the story, the geometry is, and a stiffener or backing makes way more sense than loosening your stitches.