There is a letter from Charles Longuett-Higgins, son of Cowper's friend, which settles this question quite conclusively:

I can with certainty inform you that the Poet himself, and all his immediate relatives and friends, used to pronounce his name as if it was spelt Cooper - that is without the w.
My dear Father and grandfather, who were among his most intimate friends, the whole time of his living at Weston Underwood and Olney, knew well this to have been the case. I have myself heard my father say so very many times.