Before each session of nursing, pump out some milk first, either using a pump or even hand express. This helps with the let down and after the let fown baby should be able to drink faster. Stop all bottles. You mentioned fast let downs, how often do you pump and is your milk flow very good? If let down too fast, hard for baby to drink. Crying could be due to his hunger, after which he gets frustrated. Best to get this solved asap, we do not want baby to be discouraged.
Hang in mummy!!