Billy Gunn has been simply going by “Billy” in AEW lately. The most likely reason is that the WWE reached out with a cease and desist on the name. As a team, they have been calling themselves the Gunn Club.
Although Billy Gunn hasn’t been using his full name, his son Austin Gunn has. This also allows AEW to call them “Billy and Austin Gunn” so that the Gunn name appears in the same title as Billy’s. Cody Rhodes did something similar when he first arrived outside of WWE by pairing himself with his wife, so that they could be announced as Cody and Brandy Rhodes.