Canon s95 amateur using full auto looking for something faster for fast moving toddler. Ideas?

Currently, we have the Canon s95 which my wife and I love using to take photos and videos of our 16 month old son with. It's compact, simple, and we manage to take great photos and videos (by our standards) that are mostly keepers in full auto mode. I know most advance users here will use A, P or full manual modes but for us, full auto has been consistent enough to give us the results we want (sharp, pleasing colours, contrast) with no fiddling.

Everything has been great until our son really started to move about. The problem we now face is the autofocus just can't quite capture his movements. Flash is less than ideal because for us it just kills the joy out of the photo. I purchased a Nikon d5100 a few months ago but it was just too bulky and cumbersome. It wasn't helped by the fact that my son always wanted to touch the big kit lens rather than carry on with his day to day toddler life, which wasn't an issue with the more discreet s95.

I'm looking to upgrade to a mirrorless camera using full auto mainly for 1) FASTER AF and 2) BETTER IQ for photos of our son, particularly when inside the house. Video is used a lot currently, but I'm a lot less picky with video quality than with image quality. I know I could learn more about photography, but the reality is we're a P&S kind of people albeit with good intentions. I know there's compromises, I just hope the compromises are suitable for our purposes.

I was looking to get the Olympus epm2 with the panasonic 14mm/2.5 and the Oly 45/1.8. I was thinking the RX100, but I think the AF won't be fast enough? The othe real consideration is the GF5 with built in flash that I can bounce off ceiling, but I'm not sure the image quality would be that much better than the Canon s95. Wife doesn't want to spend more unless it's considerably better (ie. faster and nicer IQ).

It doesn't need to be super compact like the s95, but not as big as the APS-C type SLRs like the Nikon d5100 or Canon rebel cameras. Budget of $1000 max.

Given our situation, what would you recommend? Would love to hear your suggestions and ideas.