TLDR: Pretty much impossible with current technology as I understand the question.
I think you are focusing on the wrong part of the problem. Turboprop vs turbofan is pocket change compared to the amount of rocket fuel required to get to orbit. Let's put some numbers to it.
I'm assuming the following mission 1) Deorbit from low earth orbit, 2) landing (potentially parachute assisted), 3) a 10,000 km range for either one very long flight or a number of short ones, utilizing vertical takeoff 4) return to orbit. Further assuming that all fuel for the entire mission is onboard at step 1.
We are going to start at the end, and work backwards.
First we need to know the mass of the vehicle. I'm assuming based on "typical science fiction dropship" that the actual vehicle weight might be similar to an F-35 fighter. Let's take the empty weight of 13,290 kg (also similar to the v-22 osprey that you mentioned). Some sci-fi movies depict a "drop ship" as having a crew of dozens, so this might actually be on the low end, but let's start here. How much rocket fuel do we need to get that kind of a vehicle back into orbit?
Assuming the delta-v required is around 11,200 m/s, assuming a sporty specific impulse of 470s (the absolute best rocket listed in this table), and utilizing the rocket equation gives the fuel required as 150,000 kg. For comparison, that's about the empty operating weight of a Boeing 777. Now, that's assuming that you start from the ground and use the rocket the whole way. Of course, if you use the jet engines to get started and launch from in-flight, you'd need a little bit less fuel, but it's not as much as you might think. Elon Musk thinks its only about a 5% savings. Let's just keep the 150,000 kg figure.
Now, we need to fly something with the weight of a B777 around a bit. For B777, a fuel load of 100,000 kg is enough to get around 10,000 km range,
So now our vehicle plus fuel is 250,000 kg. To do a vertical takeoff, you'd need a whopping 2.5 MN thrust, which in jet engine terms is around five GE9xs at max takeoff thrust (which by the way are about 10,000 kg each, so now our vehicle is way over our initially assumed weight, which means we need even more fuel!)
That's already a tremendous vehicle. And we haven't even talked about how to get this thing to survive re-entry without burning up. You'd need a massive heat shield, which would just add even more weight. I would call this impossible with current technology.
So to modify it to make it work, you'd need to do one or more of the following
- Massively cut back on the final mass. i.e. make the final thing look more like an tiny Apollo command capsule than a "typical sci-fi drop ship".
- Re-fuel on the ground before each flight and before returning to orbit.
- Give up on VTOL
- cut down on the flight range
- give up on hard sci-fi, and invent a magic rocket engine with a tremendously better specific impulse than current technology.