Maybe my earlier post was too wordy to bother with.

Let me try again. In my two main venues I've been taking a mono mix from the board into one channel and an AT3031 for room ambience iinto the other. Pretty good.

But two improvements I think I could make are:

1. to take L and R stereo from the mixer when available, and add the ambience mic to both channels.

QUESTION: Would a three-channel mixer be the best approach, or would it be better to record the ambience on a third track (in another device)?

2. to move the ambience mic up to the front on the general principle that closer to the source is better (it's currently at the rear, with the time delay corrected in post).

QUESTIONS:
a) Would a good omni be a better choice than the AT3031 cardioid in that position (up front)?
b) If so, would a good wireless omni lav good enough for the frequency content and dynamics of musical events?

A 3 to 6 channel mixer would be a good choice to have for these activities. Even if you don't use all the channels for recording, it allows you to hook up multiple mics and audition them with a couple of button clicks during practice sessions. A mixer like a Mackie 1202-VLZ or 1402-VLZ also has enough extra inputs to allow monitoring the returns from the camera in mono, which is very important if you are recording multiple sources. Since you can't undo mixing in an ambient mic, monitoring in mono is the best way to hear that something isn't right.
A lot depends on what the house is really sending out on left and right and if that is really useful for the situation. It also depends on how much work you want to do in post. Recording the ambient mic on a separate recorder is safer, but obviously much more work for sync and drift issues. One way we haven't mentioned on this board to combat drift is to use a separate video recorder or camera in VCR mode. Record the video out from your main camera and the separate mics. This will be drift-free since it's locked to the main camera's video signal. And it gives you a spare recording of the video as backup. We've probably never mentioned it because people want to use multiple cameras for multiple viewpoints, and many cameras and VCR's do less well at audio recording than pure audio recorders do.
I would definitely move your ambient mic closer to the stage.
On the whole, I don't think an omni would be a better substitute for a cardioid. Sometimes it would, but many times it wouldn't.
I also think a wireless lav would be inferior to a 3031 or 3032 mounted in the same spot. A wireless lav would however be better than a regular mic that you mounted in a bad acoustic position because of safety or practical issues with running the cables from a good acoustic location.