Think about it: You would be sending your entire Bitcoin savings into a single point of failure, which is linked to an exchange, which may or not have your personal data, including IP, real name, real address, picture of your ID if you want bigger withdraw limit...
The new BTC wallet can be any BTC wallet (ie. a software wallet on your PC, a hardware wallet, a paper wallet or an exchange etc)... where you send it is up to you. The point is to put your BTC on a different seed/keys, so that should the BCC wallet you choose to use be "Bad"™ and steal your seed/keys... the thief will only be able to access your BCC, as your BTC is already some place else.
It doesn't matter if its BTC or BCC, the point is, you would be sending your entire Bitcoin savings (in whichever blockchain) and the exchange would know, because generally exchanges don't create more than 1 deposit address by account, which means, you would be telling them (I got all of these coins, all of these addresses are mine).
Also with some simple blockchain inspection, they can see that you moved your coins from wallet A to wallet B after the split, so they can also find your BTC savings.
Bitcoin's lack of fungibility: an huge problem.
I'm with HCP in that I'm not seeing the problem. My primary objection is you don't have to use one single exchange. There are loads of them. Plus with the amount of volume exchanges generally see, any addresses you might use are pretty much the equivalent of white noise to them. I doubt anyone is looking that closely. But if you still want to be on the safe side, you could run your funds through a mixer while dividing them into any fraction you want and send those fractions to any number of addresses you wish to freshly generate, then each address can deposit funds to any number of different exchanges. So even if one exchange did fail, you'd still have funds on others. You would only be sending your entire balance to a single exchange if you freely choose to do so. It's not a mandatory requirement unless you've been reading some really weird instructions that are definitely wrong. It's entirely up you you how far you want to go to protect your privacy and security.
I'm assuming the lack of a mixer for BCC/BCH isn't an issue unless you're actually worried about those being traced, which I somewhat doubt if your goal is to dump them. All the exchange would know for sure is that at some point, that original address you're worried about held all those coins, but then they started moving and the exchange wouldn't know who has them now. And again, upon leaving the exchange with your newly acquired BTC, you can utilise a mixer before sending the funds back to your desired address (obviously different to the starting one).