If FaceTime is implemented the way the parent comment mentions (which it very well may not) I don't see why you could make a client that performs a proper key exchange to set up an end-to-end call and simply pretend grab the video instead of hiding it behind the "dailing" screen?
"The initial FaceTime connection is made through Apple server infrastructure that relays data packets between the users’ registered devices. Using APNs notifications and Session Traversal Utilities for NAT (STUN) messages over the relayed connection, the devices verify their identity certificates and establish a shared secret for each session. The shared secret is used to derive session keys for media channels streamed via the Secure Real-time Transport Protocol (SRTP). SRTP packets are encrypted using AES-256 in Counter Mode and HMAC-SHA1. Subsequent to the initial connection and security setup, FaceTime uses STUN and Internet Connectivity Establishment (ICE) to establish a peer-to-peer connection between devices, if possible."
I'm not doubting whether FaceTime is end-to-end encrypted, I'm not sure whether FaceTime sends data before the call is accepted for speculation reasons.
