Heat the vinegar and sugar together in a non-reactive saucepan. Simmer gently, stirring until the sugar has dissolved. Stir in the salt, set aside and leave to cool.
In a large stainless steel or glass bowl, mix the carrot, red onion, and radish together, then pour the cooled vinegar over the top. Set aside, covered, for 4 hours.
In a small bowl, combine the mayonnaise with the lemon juice, zest, and hot sauce (if using).
Toast the brioche rolls lightly.
Check the crab meat for any pieces of broken shell then mix with the mayo and chives. Season to taste with salt and pepper.
Put a couple of salad leaves into each roll and pile the crab meat on top. Dress with pickles and serve.