Is there a word for a group doing a cover of an originally instrumental tone, where the vocalist carefully matches the original instrumental solos' pitch and rhythm?
This style is called vocalese. From Wikipedia:
Vocalese is a style or musical genre of jazz singing wherein words are sung to melodies that were originally part of an all-instrumental composition or improvisation. Whereas scat singing uses improvised nonsense syllables, such as "bap ba dee dot bwee dee" in solos, vocalese uses lyrics, either improvised or written and set to pre-existing instrumental solos, sometimes in the form of a tribute to the original instrumentalist. The word "vocalese" is a play on the musical term "vocalise" and the suffix "-ese", meant to indicate a sort of language.