zeeoo,
Cold jets with a piston engine and added compressor?
How could a piston engine compete with a turbine that has a high power to weight ratio and a built in compressor?
A better plan might be installing turbines at the tip. Hiller was working on that also. R/C model people are building small turbojets, not sure if they have gotten to building turbofans, that could be interesting.
Electric turbofans is another idea, used in the R/C world. A small battery could provide a surge of power for a jump takeoff.