@jaem Use Simple Trainer, as it can detect your MP character in Director Mode and save it.
To do this having installed it, go to Director Mode, select your MP Character, then open the trainer, navigate to Model Spawning, Clothes Menu, MP Model's Customization, hit Detect MP Model in Director Mode, then go back to Clothes Load/Save Menu and select a slot to save your MP Character.
After switching to Franklin in Story Mode, change the model to MP Male, select your MP character from Clothes Load/Save Menu, then open CharaterSwap's menu, select presets, hit Franklin's hash and you should be able do the normal stuff in story mode with your MP character.