You're up against it to find something small with remote.
Handlebar mounts are reasonably common and just screw into the cameras tripod mount.
But, most of the lower end compacts aren't terribly robust and will be pretty unhappy sitting on a handlebar with all the jarring, etc. Unfortunately that also applies to a lot of the higher end compacts as well.
Also sweat, sand and dust will kill a camera faster than you think.
It may be worth consider a tough camera, waterproof, drop proof, etc. Either Olympus mju 8010, mju 3000 or similar (early models than the current generation use xd as the memory cards which are painful, they are now sd) or the Panasonic FT1 or FT2.
Also most of the newer generation compacts will shoot HD video.
Sony and Samsung do offer some cameras with touch screen technology, so you could trigger the shot with that method.
The big but is, that irrespective of what you settle on, if you run it on continuously as you ride the battery life will compromise you significantly. Those bright back screens just chew the juice.
My two cents worth would be go the waterproof/tough route. If you don't then choose a camera with a folded path lightway, that way there is no telescopic lens out the front of the camera body, so more likely less sensitive to damage through vibration, but also less likely to be wiped off, or jammed in the case of misadventure.