You do not need a Company Secretary for a Singapore Branch Office. However, it is highly recommended to have one as he/she can help you to stay in compliance with statutory regulations.
Suggested Readings
Must the foreign branch share the same name as its parent company?
How do I register my Branch Office with the Accounting And Corporate Regulatory Authority (ACRA)?
What is the procedure for setting up a Branch Office in Singapore? How long does it take?
What tax incentives will I enjoy if I were to register a subsidiary company rather than a foreign branch?