You will not able to control your online wallet during that time. So, you have to hold your coin on Bitcoin Core. But I have invested all of my BTC. I have bought some Altcoin with my BTC. When this problem will solve then I will sell my coin. And again I hold BTC. Whing this period I hope, I will make a good profit from all of my Coin.
How come? Bittrex says they will comply a possible bitcoin split. They won't steal our bcc, will they? Then what's the problem really? If one exchange says it will provide bcc, it will for sure.
It is the case with most exchange sites, and some other online wallets, services. They all sent mails to us customers what will they do in case of chain splitting, and with your statement I agree if they said they will do something I don`t see any reasons why they wouldn`t do that, its in their interest to keep their customers satisfied.
Its just recommendation what is the best to do, but we all have a choice. I think many people take this too seriously, when everything pass you will see that there wasn`t a need for any worries.