If you wanted to live up to your name (lazydog) you could buy the purée instead of doing it yourself. Then you wouldn't have to do anything but add it to the carboy. I've never looked for strawberry purée though, but I'm assuming you can get it. I recently made an apricot purée. I added the 3 lbs of purée straight to the primary without any preparation and it turned out fine.
Otherwise, since you're using fresh fruit, your technique sounds like what I would do in this instance. Let us know how it turns out.