Sony Alpha Lens on BMCC

To BMCC support: which mount would be better, the EF or MFT, and I assume I'd need an adapter, of course. Is this possible? Or would neither mount work?

The Alpha lens won't work on the EF mount as the Alpha lens flange focal distance won't allow for an adaptor. So you would end up with infinity focus issues unless the adaptor had an element in it to correct this. Which would be very expensive to do it correctly or of bad image quality if done cheap.

You would need to look for a Alpha to MFT adaptor and order the MFT mount camera.

Any lens that is fully electronic would not work.

What about the Sony Sony LA-EA2 or the Sony LA-EA1 Adapter?
any experience with this?

Both of those are for a-mount to e-mount, i.e. Alpha to Nex. The BMCC is available in EF and soon apparently (finally) MFT.

What you need is a-mount to MFT. Fotodiox makes one. It seems fine on a Pansonic MFT camera.