its hard to see how bad that splinter is, you obviously have considered rounding the edges more to deal with that splinter, but if the bow is tillered and finished, that could affect the tiller.
putting rawhide on would be the quickest and also not affect the draw weight, and be the quickest.
sinew will take a lot longer to dry and then will have to retiller.
now this next point will raise a lot of contention, i follow a guy on stalkbook that builds a lot of short native bows, and has had a lot of very in depth experience with sinew- and his belief is that any bow over 55" or so , the sinew is wasted on, the limbs are just not moving enough to fully engage the sinew.that does not mean the sinew wont work, it will. its just that you are not maximising its potential, for a huge gain in mass, effort and expense( if you have to buy the sinew).
save the sinew for a project where it would work better.
personally, i would use rawhide, do you have some nice thin rawhide?
after using either thin ca or hide glue to glue the splinter down.
but, a light synthetic thread and ca glue would get you shooting in 24 hours.
i have used the thinnest serving thread i could find, and wrap it just like you do the serving on a string.- then saturate with thin ca, a few coats.
but imo rawhide will look the nicest, and unless there is reason too, i like to stop the rawhide about 8-10" from the tips, keep the weight off the tips. finish with a nice contrasting silk thread and ca wrap.
good luck mate