I cant comment on the 200 f/1.8, but I do own & use both the 200f/2 & the 300f/2.8 V2, both are amazing Lenses, both sharp, fast to focus & reasonably similar now (with the weight reduction in the 300 v2) in weight.
Of the Lenses I own (the above 2 plus the 400f/2.8 V2 & 600f/4 V2), the 300f/2.8 V2 lens is my most used Long Lens (Focal Length 200 and above), I'm continually amazed by it's sharpness & although I prefer to shoot this Lens on a Monopod, it's very hand holdable as well, as is the 200f/2. I shoot both on the 1Dx, 1DMKIV & 5DMKIII.
Only issue I have had with these Lenses has been the ridiculous requirement to have to send both Lenses (As well as the 400 & 600) back to Canon Singapore to have the Firmware upgraded, no noticeable difference with the 300/400/600 since upgrading but now the 200 works on the 5DMk3 without the Grinding noise during focus hunt it had prior to the firmware upgrade.
Either Lens is going to work for you I'm sure, my pick though would be the 300f/2.8 L V2, in my opinion the best Lens I own, bar none.