You'll run into a parallex problem trying to shoot right handed using your left eye. I've tried this, (my right eye was damaged at age 7) shooting left handed will not only decrease the time it will take to get your good left eye to be your dominant eye, but will also avoid the parallex problem, IMHO.
Not to mention you'll now have your stronger right hand/arm for your support hand, which is't such a bad thing.
My 2cents,
Mike
PS - Anything which requires the use of one hand and has not been attempted before, I use my left (I was born right-handed). What can it hurt to know how not only to use your off-hand for multible tasks, but increase the use of the right side of your noggin.