I think it's good that you're showing ingenuity, but it would be bound to be a large, heavy, expensive piece of equipment. Also, turbine engines are subject to a great deal of delayed response. Having a positive displacement in the middle vs burn cans may help it to a degree, but the additional weight and bearing surfaces in the rotating assembly would negate the primary reason people use compound turbos, response time.
I'm not saying it's impossible, but you would get into diminishing returns real quick.