What do you mean by "slow"? Does it take like 3min to get to temperature? If so then there's probably an issue with the mosfet on the controller board.
Otherwise what you described looks like a matter of thermal mass. Even if two tips are shaped the same way, the one that's heavier or bulkier usually stores more heat and can deal with bigger instantaneous heat loss from coming in contact with a big component or large ground. Also, counterfeit T12 tips from China tend to not be as well made than the original Hakko's (less copper, thin plating, poor bonding with heating element). Even with 1KW power supply and controller, if the tip isn't physically capable of delivering the heat into the board, you might as well just be working with a 20W iron.
So, at the end of the day, cost cutting and crude manufacturing can fail to make decent T12 cartridges much easier than it can fail to make plain 936 tips. If there's a T12 type you really like and use often, you should buy that one genuine from Hakko.