I see the rhyme, but no one, and I do mean NO-ONE, ever says "We couldn't care fewer." (Obviously, it should be "...care LESS")

It should be "whoever spoke." The use of the verb "spoke" makes "whoever" the object of the sentence, not the subject.

Whomever should be whoever. If one rewrites this segment of the lyrics as the sentence "The echo of whoever spoke was faintly bouncing around the room," one can see that there are two verbs, "spoke" and "bouncing" (or "was bouncing" if one includes the auxiliary verb). Every verb needs a subject. In this sentence, the subject of "bouncing" is "echo," and the subject of "spoke" is "whoever." One can not use the object form "whomever" in the subject position. Perhaps the misuse was for the purpose of artistic expression, but if the misuse was an unintentional mistake, I believe the error may be due to the preposition "of." While it is true that every preposition must be followed by an object, an entire clause may serve as an object. If a clause follows a preposition, by definition, there must be a verb in that clause (as a clause must contain a subject and verb by definition). In this case, the entire clause "whoever spoke" serves as the object of the preposition "of." And as mentioned before, as all verbs need a subject, so does the verb "spoke" in our case; "whoever" is that subject.