There are few renowned and popular exchanges which are no problem to exchange crypto big amounts such as- Binance, Kucoin, Kraken, Coinbase, Houbi, Hitbtc etc. But every exchange has few rules and regulation such as fix withdraw amount, kyc process etc.
Yes, they can turn on to your complains if something goes unusual with your transaction. They may require KYC but that's for sure that they will take care of your transaction. I refrain using your suggestion with Hitbtc, there's many complains about them.
For example.
HitBTC scam!Their thread also has a lot of complains.
HitBTC.com - The most advanced cryptocurrency exchange