There is not an easy way to do this I’m afraid.
Depending on the token and gas costs you either send enough funds to submit an unlimited approval for that token to the master account or enough funds to to directly transfer it out from that account.
What I’m assuming cex’s do is bundle the sending of funds and transferring the tokens in 1 request so it would fail completely if one of the actions fail and you won’t pay a gas fee as you only pay for successful transactions with parties like flashbots. These can be calculated exactly to the Wei so it doesn’t waste gas and are all batched to be executed on the lower prices of a specific time period. They probably have deals with parties like flashings/MeV parties to do this on the cheap.
Also to save on gas you can probably use the approve method and reuse the deposit addresses, making sure to reuse addresses that have already executed unlimited approvals. People will have to request a deposit address just like on binance, except of always having a fixed address you just “reserve” one of the preapproved ones if available.