One could argue that Logue could have been portrayed in a less showier fashion, and yes perhaps he could have been, but really in the way he is written in this film it seems unlikely. He is character who is always making sly remarks, and purposely acting a bit theatrical. After all there automatically is the scene where he is entertaining his sons through his acting, it would be very difficult to act that scene without doing it in a theatrical way.
Rush takes the theatrical style, and he does do it with gusto. Rush most certainly does know what he is doing here, and he does do it well. His whole point is to purposely provoke the King, and in a way manipulate the King to help him speak more properly. Rush handles this well, being quite flamboyant in his way of mocking the position, and institutions of the royals, allowing for entertaining performance when he does this. His reactions to Firth's actions are always well timed, and well done even though they are not exactly subtle, but entertaining which is what he was going for. Rush knows how to use his manner, and his voice just right in a performance like this to this effect.
Rush does have more quiet scenes when Logue tries to find the psychological problems behind the King's stammer or more specifically when he is giving the ques to the King when he makes his climatic speech. Rush somehow remains still always very controlling, even in this quieter scenes. Again what he does are reactions to Firth, which again he does well, and I did believe his ability to manipulate the King, because of Rush's special manner of delivery. A showy role for a showy performance, I could easily see someone really disliking this self-aware performance, but I thought it did work out for the character.