Would speech therapy help a child with selective mutism, if he already speaks fine when he wants to speak?

No: No, speech therapy helps to correct functional problems with speech. Selective mutism is psychological/developmental in origin, and should be addressed as such.

Maybe: A creative and interested speech/language pathologist may be useful in this setting, just as they could for performance anxiety in public speaking. Routine "physical therapy" type of speech therapy wouldn't be helpful.