Drones cannot use skills of other classes, or player skills that are granted by items they have equipped (unless they've made some deep change in the latter).
What Tactical Mode likely refers to is skills that count as being used by the Drone. That should be Gyroblade, since Global/Steam made it passive. It could also theoretically include Medpak and Shield Generator, if they still include activation pulses - in which case it should not lower the "command" cooldown, since those are player skills, but it should increase the frequency of pulses *if* those are considered Drone skills.
But in all cases, it really depends on how they set those skills up as clarity wasn't their strongest suit. If Retrainers are readily available, the best way to make sure should be to test how Gyroblade and the Drone "commands" interact with Tactical Mode.
