... it is a mirrorless, but with the kit lens it is lighter than the LX7 I believe and contains a micro four thirds sensor. The problem is that the lens is relatively slow compared to the LX7... but it is of good quality according to reviews
I suggest against the RX100 as Sony is not good in supporting HDR or bracketing (but check on status of this for the model you want)... plus the GM1 has a larger sensor and if you get bored, you can switch the lens. The problem with the Panasonic Lumix GM1 is that it has no in body stabilisation - you need lenses with it unless you use it on a tripod.
I am looking at this for the same reason, as a back up camera, and after having oscillated every which way, I think I will get the GM1... IF a stabilised fast pancake prime is released for it. Otherwise I am waiting for the LX9... if it ever appears (I have my doubts that a LX9 with a larger sensor and fast lens is possible).
If you are not attached to totally portable and budget is not a limit, check out the Olympus E-M1 and Fuji X-T1 as backup cameras.