Timmbits: The short answer (to the Title question) is:
Both have a similar sensor size (actually used - we don't concern ourselves with unused parts), so that should not be a strong factor in our decision.
G7 comes with a kit lens, that is 14-42mm f3.5-5.6, this is equivalent to 28-84mm f/7-11 in FF terms. actual (physical) maximum opening is 4mm-7.5mm in diameter. costs $800
LX100 comes with a lens that is 10.9-32mm f1.7-2.8, this is equivalent to 24-75mm f/4-6 in FF termsactual (physical) maximum opening is 6.4mm-12.1mm in diameter.costs $900 (and hasn't yet come down in price - it was just announced)
So the simple answer is, you are getting a much brighter (faster) lens (and more glass despite the 10% smaller sensor) with the LX100, for about the same money.
Although you can't change the LX100's lens, at this price point, you aren't going to do much better no matter what you add to an MFT body. The more interesting lens affords you more flexibility than the GX7's kit lens.
I think the LX100 would satisfy a lot of shooters' needs. Personally I want the ability to go both wider and longer than 24-75mm so it wouldn't be a replacement for my GH4. OTOH, I'd really love to have one for walkaround and travel. Maybe when the price comes down a couple hundred bucks.
Archiver: I love this upgrade, but I'm wondering about what is NOT being said. HDMI output has a maximum of 4:2:2 10bit quality, but what about the in-camera files themselves? And what of useable dynamic range?
While the 4k output and 200mbps and 100mbps ratings should be excellent, what about the bit depth and dynamic range of the in-camera files? Surely Panasonic would make something of quantifiable advances in those areas?
I could be mistaken but I don't think you need the dock to record 4.2.2 out. I believe you can also use the built-in mini-HDMI port to an external recorder.
HaroldC3: 12fps is very impressive. Are we verging on the ability to shoot sports with mirrorless?
Consider that you can shoot at 30fps for hours on end at about the same resolution as older DSLRs like my old Canon 20D! That's still plenty of resolution for many purposes.