You can adjust the clipping distance of the camera to solve this, but if I were you, I would have added the skydome to a background scene instead. Then I would make the camera in that scene copy the rotation of the camera in the main scene. This can be done by a simple script, and even by only logic bricks.
Put your skydome inside the new scene and scale it down so that the camera is just inside the sphere.
Go back into the main scene and select the camera. Go into the Logic Brick editor and add an Always sensor connected to an AND controller and connect that to an overlay scene actuator. In the input put the name of your skydome scene.
After that you should make a script for those two cameras where you copy the rotation from one to another. If your using blender 2.49 you need to store the rotation in a 3v3 Matrix so that you can add that to the skydome camera orientation. In 2.5 I think you have to use Quaternions instead of a Rotation Matrix.
You should spend some time reading about python and the blender API before you try to assemble this script. I’m sorry that I can’t make an example for you now because I’m on vacation and I don’t have my computer here