Strap buttons would definitely be the way to go, from my point of view. That or sitting down. I have a custom Keith Young walnut and spruce 'harp that is as beautiful and autoharp as you'll find, and it's got brass strap buttons which enhance rather than detract from the beauty of the instrument. I'm having Keith build me a new 'harp - all koa - which should be spectacular in every way, and again I'm having strap buttons put in. They should be standard for all professional players.
The buttons will have no effect whatsoever on the sound because they'll be attached to the frame. Autoharps have a massive amount of string pressure (aproximately 2000 lbs!) and therefore need to be built around an especially thick frame. Put your buttons at the top of the harmonic curve (above the chord bars) and below the chord bars by the low F string, at the corner of the instrument, and you should be well balanced and good to go!