Baryta Prestige goes through the Pro-2000 (just like a 4000, but narrower) just fine - it is one of my favorite papers to use on mine. I also concur that it doesn't much care about being switched off for quite a while (months) - it may gurgle away for 15-20 minutes when coming back on from an extended sleep, but it prints right away, without so much as a manual nozzle check. If you do blow a head, it's a $500 5 minute replacement, unlike the Epsons where a head replacement is $1500 including the service call.
You can't run things that won't roll at all through the Pro series since the paper paths all have a ~180 degree turn. By contrast, the Epsons will print on anything including inflexible sheets of plastic or metal (the limit is 1.5mm thick, but it doesn't have to be flexible at all). I have a roll of silk on a backing paper that I'm trying to decide whether I trust on the Pro-2000 (I'm a little worried that the curve might cause it to detach from the backing). Interestingly, Canon claims to handle heavier roll media than Epson (Canon up to 0.8mm, Epson only up to 0.5mm). Looking at the paper paths, that's not intuitive - the Epsons have very little curve other than the roll itself - the paper feeds down from the top, passes the head a few degrees off vertical and comes out the bottom - essentially a straight, nearly vertical path (the cut sheet feed is truly straight, while the roll feed has a slight curve coming off the roll). The Canon feeds in the bottom front, makes a u-turn, passes the head horizontally and feeds out above where it came in.
The Epson paper path is clearly better, but it also leads to Epson's tricky pressurized ink system. Canon has a much simpler ink path with many fewer clogs in return for a slightly convoluted paper path. HP and the older 24" Canons got to a nearly straight, mostly horizontal paper path without putting the head at an odd angle that requires the tricky pressurized ink system, but that means the roll holder is behind the printer (the Canons put the roll holder at the top rear, accepting a slight curve, while the HP path is straighter, but the roll is more directly behind the printer). Hardly the best place to load paper... All three have advantages and disadvantages.