Each 30l barrel pours 90 glasses. if you ask for 3 barrels, that's 270 glasses. if your guests drink beer, they will most likely have 2 to 3 glasses each for the entire dinner. that's enough for about 90 guests. you are inviting 200 guests, so 3 barrels are enough for less than 50% of your guests.
i think there's no harm asking for more first. note that some hotels provide free flow beer. if later on you think that your guests will not finish 3 barrels, you can then exchange the 3rd barrel for something else.
by the way, my package says additional barrel at $750++. not sure why the difference
but i get your point on the free wine per table thingy. worth trying also although i think i've not heard many couples who got that.
don't worry about being a lousy negotiator. just ask for the list of items in a nice manner. i initially asked for free flow beer. the coordinator said "that might be difficult but let me see what i can do" and the very next day i was told i can get 2 additional barrels.
try for other things like additional night stay. also note that most hotels provide free food tasting but not mandarin court. i forgot how much they charge for food tasting.