I'm not sure if sprinting will help your metabolism any more than weight training, but an interesting idea.
I think if when you are doing your three weeks of weight training, you workout hard enough and with enough volume to cause a slight over-training effect (performance drops slightly) that you will need an intensification phase and sprinting may actually do the trick. I'd do the sprinting for only a week though.
Neither I or anybody I've trained have tried it so I cannot give an experienced answer, but I can say that I'm intrigued. If you do it, and it works, let us know.
Worst case scenario is that it doesn't work as well as you thought. It won't hurt to try it.
As I was proofing this I was thinking more and more that it may be worth my trying it. I hope you give it a shot.