I’ve always thought the rate of flashing was dependent on the amperage load found in the circuit. That is why old school flashers fail to properly perform when all the bulbs are replaced with LED. With this in mind I have no idea why hitting the brakes could increase the flasher load; The brake circuit and turn signal circuit are not by design connected at any point
do you have an old-school flasher or a newer LED capable flasher? Not sure it makes a difference just trying to gather information
Larry