What does the phrase from the FAA, "clear to maneuver" mean?
That isn't standard FAA phraseology; at least, it isn't anywhere in the ATC orders. The closest thing is in section 9-2-20 on evasive action maneuvers. I know nothing about them, but they seem to be a military training thing:


  PHRASEOLOGY−
  CLEARED TO CONDUCT EVASIVE ACTION MANEUVER FROM (fix) TO
  (fix),
  and
  (number of miles) EITHER SIDE OF CENTERLINE,
  and
  MAINTAIN
  (altitude) THROUGH (altitude),
  and
  COMPLETE MANEUVER AT (fix) AT
  (altitude)


But of course both pilots and controllers use non-standard phraseology all the time, especially for unusual situations. It's certainly possible that a controller somewhere has indeed said "cleared to maneuver as requested" or something similar. Although the 'correct' phraseology would be "approved as requested" (see section 2-1-18 of the orders) for a general request.

Another possibility is that it's military phraseology, not civilian. If you can link to a source or ATC recording then you might get a better answer.