I can't find the specific amount for a complete drain and refill, but I am certain (after rebuilding 100s) of automatic transmissions, that the figure in the owners manual (3.2 quarts) for a 'drain and refill' is close to only a third of the fluid in the entire unit. This is because the torque converter generally holds about 2/3s of the fluid in an automatic, and it doesn't drain when the pan is removed. There are only three ways to remove the old fluid from a torque converter; remove the transmission and torque converter and drain it ( even then all the fluid doesn't come out), drill a hole in the side and let it drain and then tap the hole and install a drain plug ( a common practice for years in trans shops, if you know what you are doing), or use a flushing system to gradually replace the old fluid with new.
Since the first option is VERY expensive, it is normally only done when the trans has to be remove for an overhaul. The second option only works if you know exactly where to drill the hole in the converter shell so as not to damage any internal components.
So that leaves the third option, a flush. And a flush is not the most efficient method, but the easiest. In simple terms, most flushing machines use the two cooling lines from the trans to the radiator. One is the pressure side to the radiator, the other is the return. With a flush machine, the cooling lines are removed, and two lines from the machine are connected. with the trans engaged, new fluid is pumped into the return line, and the old fluid is collected from the pressure side. Because this is not a completely linear circuit in and out of the trans, some of the new fluid will mix with the old and be pumped out in the collection side before all the old fluid is finally removed. This is the reason for needing more new fluid that a 'dry' teardown unit needs to fill. I have seen quotes from local shops for between $60-$125 for the flush + the fluid. So $200. or more is not out of the ballpark.