the [LX7's] zoom is only 3.8 but I know this can be extended somewhat further by reducing the settings to 5MP which will probably provide enough zoom whilst still providing a good image. At least better than that from a longer zoom P&S like the Canon mentioned earlier and the TZ30?
Again, anytime you use digital zoom/crop mode, there will be an IQ drop. Maybe it will help to think of it this way: you already agree that smaller sensors generally have poorer IQ than larger sensors; since crop modes work by use a smaller area of the sensor, they usually suffer from poorer IQ as well, just like using a physically smaller sensor would. The crop mode is just limiting the sensor size electronically instead of physically.
As to how this would compare to the TZ30 (or any other camera), first make sure that you are doing an apples to apples comparison. e.g. at 5MP the LX7 is still limited to a 35mm equiv. focal length of ~130mm, whereas the TZ30 goes all the way to 480mm (35mm equiv.). Further, you need to compare the LX7's 5MP image to a 14MP TZ30 image that has been downscaled to 5MP. In such a comparison, I don't think the LX7's image would be better than an image captured on a TZ30 at a similar focal length and then downscaled to a matching resolution.
Of course the LX7 would not just be used for this particular event and seems to provide a good all round experience but my main question is if you feel my thinking is sound as far as using the LX7 for the use mentioned above of the school performance.
If not please advise alternatives.
... we would prefer a pocketable backup that can perform in a variety of situations and provide good IQ.
The problem you're going to run into is discussed in the two threads I linked to previously. i.e. with current technology, physics makes it impossible to have a pocketable, bright telephoto lens with an image circle that covers a "large" (1" or larger) sensor.
As I said in the other thread, I completely understand the desire to have a pocketable backup to a larger system camera--this is the exact reason I bought an RX100--but unfortunately, the situation you specify above (basically, low light action at a distance) is the one situation where no pocketable camera can do well.
Given that, and also now taking your budget into consideration, the best alternative I can think of is adding an Olympus 45mm f1.8 lens to your G2. No, it's not pocketable (because, again, no bright, telephoto, large sensor/camera lens can be), but it is:
1) roughly the same price (maybe even a little cheaper).
2) the same 90mm (35mm equiv.) Field of View as the LX7 zoomed all the way in, with a similar (actually slightly faster) aperture at that focal length. (f1.8 vs f2.3)
3) similar (actually slightly higher) 12MP resolution, so you can still crop down to 5MP in post if you need/want to.
4) using the G2's much larger 4/3 sensor, instead of the LX7's 1/1.7" sensor.
My 2¢ - Mark