If your account has a converted feature from bitcoin to altcoin, you can select the other coins with a low transaction fee. But if it's not, I guess you can not avoid the high transaction fees and only wait for a while until the fees can reduce a lot. It seems your account needs to upgrade to have more features from that exchange.
Can be alternative way, for other people too. Because blockchain allow us to swap if we do KYC in their site. Because if too many inputs which is usually by old users, blockchain is used to save coin from faucets or anything which can send small amount of money. And OP look like have much money because bitcoin price increased .