In Bermuda lawns it is typical to have bare spots in area of shade. Bermuda loves the hot sun and lots of humidity but hardly grows in the shade. If I had the issue you describe I would first investigate Creeping Red Fescue for seeding the area this fall. Plan to keep the area damp to insure germination and aid the grass to mature and develop rhizomes. The texture of the grass is a nice complement to Bermuda.