There is more than one way to do everything, so I hope you get other methods suggested.
I’d cut the groove using a graver or ball bur that matches the diameter of the gold wire. Anneal then bend the end of the wire to follow about 25% of the circumference of the ring in the groove. Hold the wire freehand or use a third hand and solder the first 8mm or so to the ring (after coating with boric acid and alcohol). While soldering you can usually pull down on the wire to seat it in the groove as you work your way around and solder as you go. If not, stop and clean it then bend and fit the wire in place. When you’ve got 25% left to go, cut it to length, push into the groove and solder.
You can try to make a complete ring of wire and snap it into the groove and clamp with spring tweeers. But it’s hard to avoid a spot or two that’s not down flush in your groove. I find it easier to do as I described. I do it this way when repairing worn channels as well, if not using the laser.