There a number of problems with the idea of transfering from a DIII school to a DI school. Assuming that you will be good enough to pitch at the larger level school, you will need to let your present coach know about your intentions. You will not get "recruited" to play at another four year school while attending a four year school. A coach will not want to contact you, you will need to contact him. When you do contact another coach, he will most likely contact your present coach and find out why you are wanting to leave his program. Generally speaking, this will not sit too well with your present coach.
It would probably be an easier route going to a junior college, proving yourself there, and trying to get to the highest level possible. This way, if the talent is there, you will get recruited to go play at a higher level.
I would suggest talking to your pitching coach on this and getting his opinion. I have obviously not seen you pitch, but you might also want to go and watch some juco games in your area and see if you think you could pitch at that level. Remember, although you said you will be the ace of your team's staff, it is still a tremendous jump from DIII to the DI level in baseball. If you are thinking about playing professional baseball, there have been many DIII players who get their chance. If the talent is there, you will get yours also.