Ah, I guess one of us should have asked what style of neck joint it was before assuming

sorry about that.
Well, I've not done a Spanish heal before but my guess is that you DO NOT take it apart.
I would find some mahogany (looks like mahogany? use some off cuts from the neck) and try as best as you can to match the grain lines of the heal of the neck and make a sliver the fits those gaps (it's on both sides right?). Now, the sliver does not need to go all the way in to the neck, basically it could be about 1/8-1/4" thick, but I would make it thicker to make sure I got the profile of the gap right. Than I'd glue it in place with HHG or fish glue (they will leave less of a glue line). Maybe even make it a wedge type fit and tap (I SAID TAP!) it in the gap. Don't drive it in or you will change the angle of the neck (or will you? maybe not with a spanish heal).
Let the glue dry overnight.
If you have some of the sliver sticking out just use a flush cut saw to cut through most of it, than switch to an exacto blade to score through the rest. Try not to touch the sides of course.
That's probably what I would do.
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