Correct gauge and jacket type for feeder wire to subpanel?
The 83% derate only applies if it is all the service to a single dwelling unit.  With 75A of A/C and two dryers, that sounds like something else.  So let's ignore the 83% derate. 

Suppose we use 4/0 wire

First, a couple common errors in sizing voltage drop are calculating ampacity on breaker trip - you should actually calculate it on practical daily load -- and presuming 3.000% is a hard number. (the voltage drop calcs online will hurt you there; they treat 3.01% as cause for a wire size bump!) 3.0% is a nonsense number not mentioned anywhere in Code. What the Code really wants is less than 8% from meter to final outlet. 

4/0 (aluminum, of course) wire, indeed, has a rating of 180 amps.  You will need to breaker this run for 180--- Oh, snap! Nobody makes a 180A breaker for that panel! You'll have go upward to the next breaker size available. It's 200A.