I had the same dilemma. A used Nex C3 from KEH solved it for me
I thought about the K-01, the native (read: auto aperture) support of K lenses is something pretty good, something the NEX doesn't do.
I planned to use the NEX mainly as a platform for vintage lenses that don't have a digital platform available: OM, Leica screw... M42... Minolta MD...
The issue with the NEX is that the lens to camera ratio is way off: mostly lens, very little camera. With the adapters the package isn't small.
Mine came with a kit Sony zoom which is fine enough. The tilting LCD is helpful, and almost as essential as I thought it might be.
I do miss the shake reduction. When you have to hold the camera away from you, there's bound to be more shake than if you held it against your head, or belly.