I don't think anyone can say with any certainty that all badges worn in the Brodrick cap were slidered, that's for sure.
I believe that the slider (Long) was brought in for the Slouch hat along with a brass socket on the inside to hold it in place.
I think badges worn on the Brodrick were both slidered and lugged and in my opinion probably more lugged, to use up the old stock of lugged KC era badges.
Neil