I would feel like something's missing as well, if I had a Strat without a whammy bar.
So far as I know, Floyd Rose tremolo units need the same rear cavity routing that a regular Strat tremolo does, while needing an even deeper routing than a regular Strat tremolo does on the top. A Kahler unit, which I'm not sure is even made anymore, doesn't need a rear cavity for all the springs, but still needs a routing in the top. I've heard that a Kahler unit does not sustain nearly as well as a Floyd Rose or a Strat tremolo. I don't know this for a fact, but I heard that a number of times back in the 80s, though who knows whether it was actually the case, or just an oft repeated myth. I don't know.
It might be cheaper to either get a new licensed Strat body that's routed for tremolo and buy a Fender tremolo unit, than to have the body routed out and the unit installed. I don't know. You might also consider trading or selling the hardtail Strat and buying one with a tremolo. It's up to you.
I had a strat a while back, and I always thought the vibrato was annoying, so I can dig the hardtail. Strats aren't really my thing though, if I was going to get another Fender it would be a mustang or a jaguar.