I'm still deciding between the Canon XF105 and BMCC for an upcoming project which will be a mix of green screen work and street interviews. The 4:4:4 Raw footage (for easier keying) is what's attracting me to the BMCC. Assuming I go down this path, which mount and lens would be ideal for green screen and street interviews?
My research so far points to MFT as having more options available and a better overal choice, but for street interviews I'm leaning toward EF (for IS stability).
Either way, I'm starting from scratch. No lenses, no body, etc... For green screen, I will be approx. 4-6 feet from the talent (space is pretty small) and teleprompter will be mounted as well. Too many choices...
Shooting with the BMCC in RAW for chromakey work will definitely give you a huge edge (no pun intended) against any other camera that uses a plethora of compression algorithms to record its footage to media. I've often said that's it's a valid decision to buy a BMCC solely as a visual effects camera for chromakey work. Not many cameras can key individual hair fibers like the BMCC can. That's impressive.
If you want some of the sharpest lenses available, then using Canon L primes is a good solution. Because they're photo still lenses, they do lack in the mechanical functionality department as compared to a true cinema prime lens, but if you're mostly locked on a chromakey set with static talking heads, then having the mechanical benefits of a cine lens may be a moot point for you, where the cost of photo lenses are much cheaper without losing image quality.
If you want IS for street use, then the Canon/Sigma mount EF/EF-S lenses are a good option as you suggest, but his is where a mechanical cine style lens does show its benefits as well, especially in regards to changing the aperture. For that, you might want to look at the Rokinon fully manual cine primes, which can be used on all four BMD cameras. Unfortunately they don't have any manual cine zooms yet. For zooms, either a manual Nikon F mount can work (on all four BMD cameras), or the original Canon/Sigma mount idea is a good option (works only on EF mount BMD cameras).