@Rosenberg,
There are a lot of factors to consider in whether dragoons should be "battle cavalry" or not.
A key factor is the status of their horses? For example, the horses for the French dragoons in the Peninsula were often in poor shape due to the lack of fodder and their use in a reconnaissance / screening role. In another example, I read that Exelmans complained about his dragoons being used in the "light cavalry" role as it would degrade their capability for use on the battlefield (i.e., as "battle cavalry").
Then there's cases like the Spanish dragoons who were converted to light dragoons and then back to dragoons. Even ignoring their chronic lack of horses, should a dragoon unit convert from light dragoons to "medium" dragoons instantly be considered "battle cavalry".
And…of course, there's lots of light cavalry who were kept in a reserve role until they were on the battlefield and who had superior mounts. They merit "battle cavalry" status, which just goes to show that the question doesn't reduce to a simple classification.
Of course, one could take the view that they are "battle cavalry" unless other factors suggest otherwise.
Just some thoughts that as to whether or not all "medium" dragoons should be "battle cavalry".