Destroyer's trait:
RELENTLESS ENGINE OF DESTRUCTION: Destroyer can't use the Carry ability. At the beginning of your turn, Destroyer may be placed into an adjacent square.

Space Gem's resource side:
Character can use Phasing/Teleport and the Carry ability. Modify character's speed value by +2.

Does this allow Destroyer to carry, or does his trait supersede the resource?

It doesn't allow him to carry. His trait isn't specifically stopping him from using the one he has innately. It's stopping him from using it period.

Alright, that's what I thought, but I wanted to double check before I missed out on some sweet Latverian Prometheus action.