As Chris pointed out, there have been several koryu kenjutsu schools that included shiai in their kata-- Jikishinkage Ryu, Maniwa Nen Ryu, various Itto Ryu factions, etc. All of these schools used fukuro shinai of one form or another, though I've heard that at least one school originally used bokken (they traded their bokken in for fukuro shinai just as soon as they found out that someone had invented the fukuro shinai). In some of his writings Ellis Amdur has described doing freestyle sparring with bokken (pulling the blows at the last moment) and shinai during his Araki Ryu training, going up against opponents armed with all sorts of practice weapons (kusarigama, chigiriki, etc).
There are several other schools that don't seem to have ever used sparring in any format. But those schools lasted-- and were popular--for centuries during a time period when being a bad swordsmen meant certain death. They must have been doing something right.
Either way, Paul, at six months in, this is only a matter of intellectual curiosity for you. Trying to spar without having a solid foundation in the techniques of the ryu would be like sitting down to do some jazz on the piano without having ever practiced the scales. Five, six years-- or perhaps many more-- from now continuous, diligent practice of kata might have burned not just the outer techniques, but the movement patterns, the sense of distance, and the timing that defines your school into your nerves and movements. At that point, there might be something for you to gain by practicing the techniques of the school in a freeform environment. In some schools (see above) that was done by sparring. In others, instructors would "break" the kata, suddenly changing attacks on the student in the middle of a kata that they have practiced a thousand times to see if they can react within the parameters of the school. I've even heard of kata breaking like that being done with live blades, though only by extremely experienced practitioners.
David Sims
"Cuius testiculos habes, habeas cardia et cerebellum." - Terry Pratchet
My opinion is, in all likelihood, worth exactly what you are paying for it.