I don’t think it’s the nicest sounding of the surnames as first names - even though I know a very lovely toddler Cooper! But Imo it’s better than super trendy or dated Jackson, Tyler, Mason etc - just not quite as nice sounding as Wyatt, Sawyer, Clark, Spencer or Sullivan if you want that type of name (which all have more classic literary or film associations - Thomas Wyatt, Tom Sawyer etc).
Actually I suppose there’s classic actor Gary Cooper though and your DS will definitely fit in with all the other boys at school - about a third of the toddlers I meet seem to have surnames as first names!