The neck angle pretty much exists to to accommodate the bridge. The Gibson Bridge is pretty tall, so quite a bit of neck angle.
Fender bridges are pretty low, so not much angle, if any at all, is necessary, so they are pretty flat.
So, if your neck angle won't accommodate your bridge, you need to change either the neck angle or the bridge. You can change the angle of the neck, but it's not simple and easy.
A shim might have fixed the problem before glue up, and as in the blog post, even the factories occasionally used shims... But now that it's glued up and not otherwise a problem it would be a pain to fix.*
The alternative is to find a lower bridge. Even if the only thing that will work is a Fender style bridge, that's not really a problem for playability and sound.
*If you do decide to take the neck off, I would recommend really studying how to remove frets and get the right tools to do it. I did not do enough of that when I pulled the frets off a bass neck to make it fretless. I had a fair amount of chip out around the frets as a result. Not as much of a problem on a fretless bass, but it did mean a LOT of sanding to get below the chip out. It would have been more serous if I had the intention of putting the frets back in!
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