broken batten

Usually if you keep the sail mega flat where the broken bit is, and get somebody else to push on the front end, it will come out without the splinters imbedding in the batten pocket.
(2 person job and slowly slowly)

Failing that, if you know anyone with a sewing machine that can do a couple of layers of denim (with care) you should be able to unpick about 6" of seam and get the bugger out then re-sew easily.
Normal cotton thread will rot, use 100% synthetic. You can use fishing rod binding (for binding rod guides on) that you get at a tackle shop for 5/8 of bugger all
Incidentally, that is really handy to have when travelling as you can sew up sail repairs pretty damn good with that and a half circle needle and a thimble. Is always in my Gnaraloo kit!

I got a long bit of stiff wire, bent one end up at 90' the thickness of the batten 3-5mm, slide it (with the hooked bit sideways)under the broken batten, turn it at 90' so the hook is behind the batten and pull