From the IGC under Joseph Rock Cooper (excerpted):
1857 Joseph and a prominent Birmingham businessman formed a new partnership, Cooper & Goodman at 32 Woodcock Street. It has been reported that Charles Henry Cooper was Joseph's son. In fact, Charles was Joseph's brother.
Charles was recorded in the 1861 census as a 42 year old gun maker living at Yew Tree Cottage, Old Chester Road, Sutton Coldfield with his wife Hannah (b.1820) and their children Charles H (b.1842 described as a gun maker) and Susan M (b.1845)
1881 census Charles described himself as a gun maker employing 114 people.
1886/87 Cooper & Goodman closed and Charles Henry Cooper started trading under his own from Woodcock Street and Excelsior Wharf.
1891 census records him as a 72 year old "living on own means". He would have been semi-retired or retired.
In about 1895 the business moved from Woodcock Street to the Cliveland Street Works, Cliveland Street. In about 1897 the business at Excelsior Wharf closed.
In about 1908 Charles Cooper's address was 67 Cliveland Street (probably the same address as previously) but in 1908 his address was also given as 77 Staniforth Street. From 1910 to 1917 when the business closed, the address was 16 Canal Street.
So it appears that Charles was the brother of Joseph Rook Cooper, and was some sort of silent partner in their business, who then went on his own when Joseph retired. Charles was 72 in 1891 so it's unlikely that he was still working in 1917 as he would have been 98. Seems more likely that his son (also Charles H. Cooper) carried on until the company closed.