Dividing all character and enemy stats by N is an interesting idea, but it would only effectively
increase the cap, and it wouldn't actually
fix the "game breaks if I go over the cap" issue (it only
delays it), and it would introduce other problems related to the resulting loss of granularity (especially in the lower ranges), and you might have to re-balance various formulas in the game.
james_c, it sounds like you're trying to "temporarily" change a "stat" to accomplish "some change in damage, speed, etc.", without violating the 255 stat caps. In order to accomplish something like this successfully, on a "global" scale (i.e. where you don't have to modify tons of scripts in the game), I'd recommend that you first read Terence Fergusson's FF7 Battle Mechanics FAQ:
https://www.gamefaqs.com/ps/197341-final-fantasy-vii/faqs/22395This explains things like "Primary Stats" vs. "Derived Stats", "Before Damage Effect" and "After Damage Effect", a hundred other factors, and all the math formulas for calculating damage, etc., in general. It will also explain the (very complicated) system that governs speed and timing.
The other thing I'd recommend looking at is to find out how the game does similar things to what you're wanting to do. For example, what happens when a character drinks a Source potion during battle, which normally raises a stat by 1? Does it permanently affect their stat after the battle has ended? How does Berserk accomplish a temporary hit% decrease? How does Haste accomplish a temporary speed increase?
You might find that you'll need to invent new stats and modify existing formulas to accomplish what you want? Not sure, depending on how creative you get.
Good luck!