Trap sense is a mechanically useless ability which actually fits flavorwise with your unarmored variant. Your player sounds like a bit of a powergamer. I don't think that you should factor trap sense into any balance calculations.
I also think that losing the natural armor bonus to AC while wearing armor stretches plausibility a bit much. His skin suddenly softens when he puts armor on? Not much you can do about this other than allowing the barbarian to wear armor normally if he takes a feat, or change the type of AC bonus.
About the balance of your original ability, I think it's pretty fair. Your AC will likely be lower in the beginning, and you won't be able to get special abilities later on, but your rage provides a scaling bonus to AC now. Adding extra fast movement into the deal kind of complicates it so you should probably just leave it as-is.