I believe that your iphone does in fact have image stabilization, although it may not be aggressive enough to be useful with bike videos. You can also stabilize video after the fact in iMovie etc.

Whatevever you use to shoot, you'll probably wind up using a Gopro mount, just because they're so ubiquitous. There are numerous options, including a bunch that have a computer mount on top and a gopro mount on the bottom. K Edge sells some, and there are a bunch of no-name vendors on Amazon with the same general idea. And if you haven't seen this thread already, there are custom 3D printed designs you can get.

I've been noodling over this problem myself and have thought about putting something together using a Gorillapod.