Yes that can work - with some difficulties and limitations. Autogiro's don't have variable blade pitch and they can fly, but they also don't have the reaction yaw issue due to driving the main rotor.
One big fixed rotor doing the heavy lifting - let's start with the device in the hover and trimmed out by the four electrical trim rotors. Then it needs to climb, since the main rotor blades are fixed pitch the rotor needs to speed up => increased yawing moment => the device starts to spin. You would need something like a tail rotor to compensate for only yaw. If the quad electric rotors are dimensioned for trimming and tilting only, reducing their rpm would not compensate for the increased torque of the main rotor.
Now we want to propel it forwards. Since the quadcopter rotors are pointing downwards the whole craft needs to be tilted, creating gyroscopic bending moments on the rotor shaft and on the blades as well if they don't have flapping hinges. Control-wise that can be accommodated of course, just apply a precessed torque on the craft so that it tilts forwards. But it would be simpler if you would have a pusher propeller installed so you don't need to tilt.
Then the craft starts to fly forwards. You must have flapping hinges now, otherwise the craft will roll over; lead-lag hinges for preventing fatigue failures. The rotor power will decrease with forward speed at first due to translational lift, decreasing the reaction yaw. We need that tail rotor again, and/or a stabilising vertical fin.
To me it sounds like this design is halfway a quadcopter and an autogiro, combining mostly the drawbacks of each design. The quadcopter has simplicity of control as its main redeeming feature, this vanishes when you add a big, engine driven rotor. As to the drawbacks of the autogiro....ehm. I cannot think of one.
The autogiro's development history is a point in case: De la Cierva's first designs used a conventional elevator and ailerons on stub wings for pitch and roll control, these were not very effective at low speeds so he made the rotor being able to tilt on a gimbal with respect to the shaft. Later as the rotors got bigger, the forces to tilt them became too high and the mechanism for cyclic blade pitch was developed. All of this in the 1920s/1930s!
So while your design seems possible, I cannot see an advantage over an autogiro with a tilt-able rotor. One big horizontal rotor for lift, one small vertical rotor for thrust.