I've played both the Barb and the bladelock. I recommend using Barbarian and instead of multi- just have him take Magic Initiate. You can have a couple of spells (take ones that don't require concentration) and go from there. Eldritch blast fills the ranged gap nicely and then maybe Greenflame blade or Booming Blade, and Armor of Agathys. Then you can proceed with full barbarian.