Jammy, it might be easier than you think.
I'm claustrophobic as well, and yet I have no problems at all diving. I did avoid a couple of the "swim thru's" in Coz, but the clarity of the water down there, makes it feel as though you are in a wide open place. As long as you are okay with putting a mask on, you should have no problems.
When I was doing my Discover Scuba in Jamaica, one guy on our boat flipped out because he was afraid of heights!
He was perfectly fine in the pool, and completed all the exercises; but when he jumped off the boat, and could see all the way to the bottom, 40 ft. below, with nothing holding him, and he paniced.