Should I do someone else's peer-review work?
No, you gain no benefit if it is you who submits the review instead of him. If you conduct a review on his behalf, you gain the advantage of his mentorship and advice. It will be good experience for you, as you will get to work through a work in progress and understand more deeply how submitted papers are evaluated. If you are concerned about credit, ask him to mention to the editor that he asked you to do the review and these are your comments. Ideally, he should ask the editor for permission before sharing the submitted manuscript with you, as it was sent to him under a condition of confidentiality.