Unfortunately changing the FOV is of little help. If you look at my screen shot, the FOV value necessary to hide the clipping would crop 80% of the image.
SnowGecko, that tutorial has already been of great use for setting up my first person cutscenes, but there's nothing there to help me with the clipping.
Thanks for the help guys but I've given up on this issue. I'm mostly just trying to move the camera out of the way of the arms whenever it is needed. It's a bit of a hack job but I have no other choice at the moment.