Fujifilm X-A1 or Sony NEX 5R/T
I have owned both NEX and Fuji cameras. I highly recommend the X-A1.
IQ: Both cameras have the same sensor, but Fuji has better lenses and a better JPEG engine.
Flash: X-A1 built-in flash can be tilted back for bounce and Fuji's flash system does a great job of balancing with ambient light. Sony's accessory port flash seems like it was an afterthought. Images taken with NEX flash have that bad look that people associate with on-camera flash.
AF: Fuji continues to improve AF and the X-A1 is one of the better performers. It still doesn't match DSLR AF, but it is just as fast as the Sony NEX cameras.
Natural Light: With the fast aperture of the 35 f/1.4 and excellent Hi-ISO performance, stabilization isn't needed. Additionally, stabilization does not help if you have people in the shot because slow shutter speeds result in blurry images, even if the people are trying to be still. When I shoot an event indoors, I set the shutter speed to 1/125 (fast enough to freeze people and eliminate camera shake) and I turn on Auto ISO; the results are great.
Shutter shock: This is not an issue with the 35 f/1.4 because it has no IS and therefore no moving elements. Shutter shock is possible with stabilized Fuji zoom lenses, but it has not really been an issue for Fuji owners; nothing like some m43 cameras/lenses.