Ike,
According to a Times law report of a civil case over the construction of the pier, the engineers and designers were Mr John Brunlees of the firm Brunlees and McKerrow of Queen Victoria Street, London. They had a contract with the Local Board in Southend who were building the pier. Messrs Erroll were employed as contractors.
The Times, Tuesday, May 12, 1891; pg. 3;
This may be starting point for you. You could track down some local newspapers of the time for further information.
Have you tried the Essex Record Office?
http://seax.essexcc.gov.uk/login.aspPhil
historyhouse.co.uk