The difference in body weight is about 200 grams. When you consider that lenses can often weigh more than the camera body... it may not be as big of a deal as you are hoping.
Another consideration (since you mentioned low light usage) ... the Canon 6D can auto-focus all the way down to EV -3. The A7 goes down to EV 0. That's a pretty noticeable loss in auto-focus performance if you switch.
I found that having a comfy strap really made the biggest difference. I didn't like carrying a heavy camera & lens very long when I had a factory strap. I picked up a Black Rapid sling-type strap and this made a huge difference. No longer mind carrying the camera all day.
I just read that the A7s will AF down to -4EV.
Sony's own spec on the a7 (from their website):
- Focus Sensitivity : EV 0 to 20 EV (at ISO100 equivalent, with F2.8 lens attached)
Canon's is -3 to 18 EV for the 6D.
The thing about the Sony cameras is that semi-transparent mirror that doesn't flip out of the way. It reflects a fair percentage of the light out of the image path in order to send it into the focus and metering sensors. You can't bounce it out of the imaging path AND deliver it to the sensor at the same time. It's estimated to be around 25-30% of the light which is about 1/2 stop (50% would be a full stop). That's the physics of it (there's no getting around the fact that they do it). But to still get good pictures (and they do) they would likely have to make up for that by boosting ISO gain... then compensate by smoothing noise and perhaps performing some edge sharpening. Of course... you can apply noise reduction and edge sharpening with any photo regardless of the camera used to take it... but those other cameras probably got all of the light delivered to the sensor.