If it is within that gaited breed's standards to allow a trot then I would trot the horse.
It is not within the breed standards of the Tennessee Walking Horse to allow a trot, so none of mine are permitted to trot when being ridden.
Actually the only one that ever did want to trot under saddle, did so because his skeleton was out of adjustment. Once the chiropractor adjusted him, he never offered to trot under saddle again.
Under saddle being the operative. Oftentimes my two that do the running walk will mimic my trotting Arab if they are behind him in the pasture.
The TWH that does the stepping pace has never trotted a day in his life - even at liberty