You could carry on with the carving and get it finished, and then if there is any split remaining by that stage that you can't carve away because of the shape needed, you could run a little superglue into the split and sand over it. The sawdust and superglue act as a filler/stabilizer. I use this method a lot when woodturning for occasional splits/cracks. Could use ordinary wood glue too I guess but usually a bit thicker and difficult to get into the split whereas super glue tends to wick in via capillary action.
If you don't want to sand the spoon and prefer a finish off the knife, you could create some sawdust from a bit of the offcut timber, press that into the split and then trickle the superglue on after to get the same result.
The glue will probably discolour/darken the timber a bit though around the edge of the split, so you might want to finish the spoon with your chosen oil first before gluing as this will limit the spread of the glue and make it easier to wipe off any surplus.
Otherwise I guess if you're artistic you might carve the split into some kind of decorative feature (but don't ask me what or how !), or put it down to experience and try a fresh piece of timber
Cheers, Paul