I would think if you line the bottom of the bento box with foam, it should be fine. Those little cameras don't weigh that much. Then you could tether the camera to your handlebars or top tube, just make sure the tether isn't so long that the camera will go into your front wheel if you do drop it!

Craft stores usually have foam rubber... in a variety of densities including a density to match the foam that lines Pelican and Halliburton cases that protect heavy camera equipment worth tens of thousands of dollars.

The way I sweat, I'd never put a camera in my jersey pocket. Once my Ziploc bag has gotten a little bit of wear and a few microscopic holes in it, I'll typically soak through whatever cash is in there. I haven't fried a phone yet <touch wood> but I keep waiting for it to happen.

Lightweight helmet cams (that can also be mounted to frames etc.) are widely available, just Google. I know the Aerostich catalog has at least one. AFAIK they only shoot video, but it's easy enough to capture still images from your video.