I'm not sure if you mean "big-ball" as a special object distinct from any "big ball." If not, I would still say "the big red ball, not the big green ball."
This is the exception that proves the rule, though, as "big" is now in fact part of the ball's identity, rather than something merely describing the ball.
Seeing this phrase (or "Begs the question") used appropriately is always a pleasant surprise.
You hear an example in Boris Johnson PM talking about "new _blue_ passport", people getting a "new passport" is a thing, but if you say a 'blue "new passport"' then people will find that a foreign turn off phrase and assume you're a non-native speaker or you're having a stroke (ie a medical problem with your brain).
I mean, yeah, if it's a noun phrase it might work, but even if we're taking about a red zorb, it's not going to be "red big ball" unless Big Ball is a trade mark or brand name or whatever.
I can't imagine any editor would let you get away with something like "the red big-ball."