Would it be wise to build and fly the 300s as my next plane. I would fly it "cautiously" and wouldn't be trying any wild aerobatics for some time. If I work with my flight teacher with the plane for a while, would this help me in being better able to handle any "bad characteristics" the 300 might have.
The other option that others have suggested is getting a "something extra" or similiar plane to be my second plane. If possible I would like to avoid this (save some money), but I don't want to greatly risk turning the 300 back into kit
I think there would definitely be some risk with the 300s. The problem is not so much the wild aerobatics, which will be easier with the Extra than with your trainer, but the landings, which will be more difficult. If you want the 300s for your second plane, I'd recommend you stay with your trainer until you are really bored with it. Stay with it until you can land it anywhere you want, when you want, you can put it down on the mains, and let the nose settle, you can fly and land in a 20 mph wind, you can land fast, you can land slow. Stay with the trainer until you can do touch and goes repeatedly and hit the center of the runway every time. Practice your dead sticks, keeping the nose down and the speed up. Stay with the trainer until you know that you won't get any better without getting a more capable plane.
Stay with your trainer until you really understand the engine. I see a lot more crashes caused when an engine fails than almost any other reason.
In summary, if you really learn everything the trainer will teach you, and you have an instructor, the chances are you'll be fine with the Extra 300s. Still, I think I would recommend leaving it on the shelf, and going with something like a GP Easy Sport. Life is long, there will be many more planes in your life.