I use the Panasonic 100-300 on Micro 4/3 cameras, and its hand-held performance is marginal at 300 mm even with IS (both the built-in IS and the camera sensor-shift of the Olympus E-M1 and E-M5). I should not expect it does any better on Blackmagic.
Once mounted on a heavy tripod with well-dampened head, optical quality is barely OK for the price, nothing more. It shows its limits especially at 300 mm.
If you can hold out a few months, I would suggest you wait for the coming Olympus 300 mm f/4 prime, which is supposed to be optically excellent (but has no built-in IS).
The Olympus 4/3 (not Micro 4/3) 50-200 mm f/2.8-3.5 SWD with 4/3 to Micro 4/3 adapter on the Olympus E-M1 it is optically far better than the Panasonic 100-300, and AF very fast. AF is instead too slow to be usable on the other Olympus and Panasonic cameras that don't have phase-detect AF in the sensor, like the E-M5. I assume that MF operation is the only option on Blackmagic cameras. With the dedicated 4/3 1.4x teleconverter, the 50-200 is still optically better than my Panasonic 100-300.