That's because you didn't set up the cullfacing on those protruding sections correctly. Anything that's poking out of a block's space should cull with the side of the block it's poking out of, and not with the direction that a given face is actually pointing.
The easiest way is the simplest: Just make sure that the pixels that would overlap are the exact same color. This is the same method used for making the overlaps on the chest not show up. If they're the exact same color, the z-fighting won't be visible even where they do overlap.
The alternative is to make sure that the bricks one one side are at a slightly different Y coordinate (offset by 0.01 or something) so that there's always one fact that's distinctly above or below the other. It's a bit more work and may not cover every single case. Plus if you ever do the top and bottom of the blocks, it'll become MUCH more complicated.
How can I make it so that transparent blocks work with the cullface? Is there a way to make them transparent or give them a property to make them alter cullface-able blocks inside the .json?
Thanks in advance!
Nope. Cullfacing is hard-coded into the game. So unfortunately it's just something that you'll have to live with.
If that's a huge issue for your bricks, consider going with the second option that I mentioned and have the protruding bricks offset by a very tiny amount. That's unfortunately all that you can do to solve issues like this.
Sorry to be the bearer of bad news.
Also, you can keep asking questions but it's suggested that you make a new thread for each one. That way someone doesn't read the original question, see the discussion is still going, and without reading the entire discussion offer you advice on that first bit which has long since been solved. If I had a dollar for every time I'd seen that happen I'd have a far nicer gaming PC.