I'm not a local but go down there about 6-7 times a year (have family in Mullion). If any locals have advice I'd love to here it.
All the climbing i've done down there has definitely needed a rope. Your best bet would be to head over to St ives (Clodgy), or Godrevy. Some good bouldering and really good online guides. Also take the family up to Carn Brea, there are realy nice views over the whole of Cornwall, a nice restaurant and some really good bouldering. The rock is f'ing sharp and eats skin. No guide for Carn Brea but locals are really helpful.
All these places are only 30-40 minutes from Mullion.