For what laid out I am going to suggest a 308. I have hunted and killed deer with a suppressed 300 Whisper, suppressed 44 mag and suppressed 308 going back 7 seasons.
Whisper
44 mag
308
The main reason I lean toward the 308 is your 200 yard requirement. You will need to have a 1 in 9 or 10 twist to shoot 200 to 220 gr bullets. Spend the money to get really good glass with target turrets. You will then be able shoot supersonic (say 168 gr) and by dialing in extra elevation shoot subsonic (say 220 gr)
Example: My 308 is zero'd with 175 smk at 250 yds @ 2550 fps. To get to a 50 yd zero for a 180 gr RN at 1050 fps I dial in 9 MOA (36 clicks)
You may be thinking about now that is too much trouble. 2 seasons ago I had my subsonic load in and zero'd. Coming across the ridge in front of me was some does and trailing them was a buck. He hung up behind a bush about 100 yds out. I decided to switch loads quickly and dialed back down the elevation. as soon as he steped out from behind the bush I killed him.
If you did not have a 200 yard requiremnt I would say a 44 mag no question it flat out smacks them when loaded to 44 special levels. A couple of examples
Here's a video I made comparing all of them
http://www.youtube.com/watch?v=Hq0N1LD937A
HTH
Pmc