I haven't been able to reproduce it yet either. I tested the save on snes9x, bsnes, and on a real Super Nintendo with an sd2snes cartridge, and I was able to win the King Behemoth boss fight and proceed to the next area in all three cases, with no crashing or any other strange behavior. So initially I was sure it must have just been a patching problem or the rom getting corrupted in some other way.
However, looking around on the ff6hacking forum, there has been discussion very recently about similar issues that can occur during the double Behemoth fight. What seems to be going on is that two of the patches included in C. V. Reynolds' bug fix compilation (which I used in this project) had coding errors which can show up under certain circumstances. One of them is supposed to address an issue where the game does not always apply the Kappa animation to characters who are under Kappa status, or revert them to normal appearance at the same time the status wears off; the other is meant to address an issue where back attacks in mid-battle do not actually change the row formation of your party as they are supposed to do. With the glitched 'fixes' applied, what can apparently happen in this specific case is that casting the Kappa spell on the first Behemoth causes the second one to show up completely invisible, apparently located in the upper right corner of the screen instead of in the middle, and casting Slow on this invisible second enemy can cause the game to crash to a black screen.
I checked valius' playthrough video of this battle, and it looks like he did indeed cast Kappa on the first Behemoth, followed by Slow on the invisible second one. So it seems he inadvertently triggered a very obscure bug, which is not present in the unmodified game. No one seems to have been aware of this or diagnosed it until just the past month or so, though apparently the exact cause on a coding level has not yet been unraveled. The only solution, thus far, has been to revert the relevant lines of code back to their original forms, or else not to use those particular patches in the first place, since they seem to be a case of the fixes causing greater problems than what they were meant to address.
The bug fix compilation patch has recently been updated to remove the problematic lines of code, so it appears I'll have to reassemble the rom again in order to completely eliminate it. Shouldn't be too hard to do, but I'll have to check over it to make sure my various edits to non-story text will still apply properly with a new version. (Hopefully I won't have to retype all the enemy and item names and descriptions again!) While I'm at it, I can also apply the updated version of the uncensored graphics compilation, which fixes a few visual errors with the Silence graphic and the mine cart escape, and perhaps also make an alternate version that doesn't have any bug fixes at all, for those who prefer their FF6 experience to be as authentically buggy as possible.
With real life getting in my way rather substantially, it has taken way too long to get back to work on updating this project, so I must apologize for keeping you all waiting so long.
(Editing to add: the character class appearing only on the status screen isn't a glitch, but a deliberate design choice by Silent Enigma, whose patch I used to lengthen space available for spell names. He felt it was unnecessary to clutter up the other screens with them, particularly since they don't really have much of an effect on anything, and I agreed.)