If you wanna do this cheap but good, you're pretty much on the right track. For around 2 grand you can get a decent camcorder with an SDI output (camcorder also allows for recording). You can run the audio from your mixer into the camera (using it as an embedder), then run an SDI cable (basically coax rated for digital signal transmission, usually dual or quad shielded, with BNC ends) for up to 300' (roughly, dependent on transmission rate and type of cable) to the overflow room. Get a projector and screen or large TV w/ HDMI in (depending on size of audience), and an SDI>HDMI converter. make sure the SDI>HDMI converter supports embedded audio if running to a TV, if not you will need an audio de-embedder to run to speakers in the room. This method should have very low latency and little if any difference in sync of audio and video. There are ways to make that even better, but more $$$. Even if the room is in earshot of the main sanctuary I wouldn't try to delay the audio to the sanctuary's audio because it would throw the sync off of the video more.
Whole setup would run around 5 or 6k, which is a fairly sizable chunk for a stand-alone, stop-gap measure. Have you talked with your churches leadership about an overall vision, and putting together a comprehensive 5 year plan for AV? Would hate for you to spend this money now, only to completely re-do everything 2 years from now because a completely different system is being put in. I know you said this is all you would ever need, but if your church is running out of space, maybe the leadership vision entails in a few years getting or building a larger building, or having several small sites connected by webcast, both of which would completely change the game from what it is now. Just something to think/talk about.