"Response: After a knelt character stands during the standing phase, kneel Lannisport Brothel to kneel that character."

My question is, can the Brothel use its reaction immediately after straightening during the standing phase, or does the simultaneous straightening of all the cards make it miss the timing (thus only allowing it to kneel someone every other turn)?

You can use it in the same phase. All cards stand simultaneously as a framwork action, then you respond to a card standing this phase in step 5 of the action window (see FAQ for more details on the timing structure).