heres a question...
if we even got the t-32 turbines./ apu etc...installed..with prop.
In AN EZE...whers does the HUGE super hot exhaust point to.
i have a friend who is tinkering with one...BUT on a RV,,,,the EXHAUST IS HUGE....on a million dollar Starship sure,,, buuut on a long?..HMMM
its probably WHY the cozy jet is a jet and not a t-prop,,,who knows?
Actually it probably comes down to the fact that there are not many turboprop pusher configurations out there. I would love to have a turboprop just for the uniqueness factor, the engines are not really the issue, there are plenty of engines out there that can just have a prop thrown on and go.... if you can find an acceptable prop. In the few searches I did though, I could not find a prop that would fit the bill for the engines. Sure you could mount a fixed pitch prop to it, but in my mind the point of having a turboprop is having a constant speed prop with beta range.
One I was looking at was a Boeing 502-6, it is already geared to run 2750rpm on the output shaft... but, it only throws out 160hp, and weighs 200lbs, which is really not a great weight/power ratio for a turbine, but would help in a CG aspect since it will be closer to the weight of a o-320 than the lighter turbines.
You also have an Allison/Rolls Royce T-63, comes in at 136lbs, and throws out 300hp... but it has 6000rpm output shaft speed... so it needs a PSRU on it, which would probably bring it close to the weight of the 502-6 with almost twice the power.
As for the exhaust... just dump it out the bottom of the airplane. You dont have to have the exhaust stack extend very far past the airframe... just make sure you use some sort of heat sheilding on the fibreglass behind it. Don't want to postcure it at 1000+ degrees in flight.

Also... once you go to a turboprop, you are probably going to spend just as much for the setup as you would for a turbofan, and get about the same fuel burn, so why not go with the turbofan which has less moving parts, less overall cost, since both a turboprop engine and a turbofan will need hot section inspections done, but the turboprop will also need prop overhauls done.