No, you do not have to change your naturalization certificate when you add your middle name.
(213) 394-4554 x0 Mr. Shusterman is a former INS Trial Attorney (1976-82) with over 35 years of immigration experience. His response to your question is general in nature, as not all the facts are known to him. You should retain an attorney experienced in immigration law to review all the facts in your case in order to receive advice specific to your case. Mr. Shusterman's statement above does not create an attorney/client relationship.
You can file N565 but I doubt the Service will agree to do that.
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You do not need to change your naturalization certificate. You may want to change your name with social security so it matches what you use for employment.
Under the guidelines set by AVVO, this response is general information only and not specific legal advice, and no attorney client relationship is formed by this response to your question.