With only about 4 hours of sun you need to make sure you select a 'shade tolerant' rose, other wise you may get the rose to grow but it may not thrive and bloom for you. I would instead of a mini, look for a small hybrid musk.

The only climbing min I know of that is supose to be shade tolerant is the red and white mini, Earthquake CL. I just got it from Nor'East so I can not swear to it.

Many of the better smaller nurseries have a search on shade tolerant roses. I know Ashdown Roses, Heirloom Roses, and Rouge Valley Roses have that search option.

Liz

it seems that you have the room for a full size climber in there. i am not aware of too many climbing minis in white; jeanne lajoie comes to mind.
i have heard that laura ford (yellow flowers) will tolerate less sun and is vigorous and fragrant as well with an eventual height to 8 feet. this rose will also set forth hips which is rare for a mini.