The EMH system is basically just an extra computer subprocessor loaded with a specific program, and connected to some holoprojectors scattered around the med bay. In all reality, a GCS probably has holoprojectors scattered around the med bay to start with, since they would be handy for displays and such. The impression that we got between VOY and the work we saw done on the EMH in DS9 is that it was a standard-issue technology equipped to all starships after it was introduced (and it would only be a trivial 'upgrade' to the med bay, not even worth calling a refit).
I think a GCS would have the best overall performance in just about any given situation encountered on the way home. Extended, independent operations in deep space are what it's DESIGNED for, and it's fully equipped for such roles (we didn't see that so much in TNG because the E-D was the flagship of the fleet, and so wouldn't have strayed far into deep space very often, so she could be on hand to 'show the flag' and such).
The Sovereign would be an interesting alternative, though. She can't go toe-to-toe with a Galaxy class - the Sovereign's only 40% the size of a Galaxy and she's a Heavy Cruiser not a Battleship, but she is Starfleet's premier Heavy Cruiser, intended to replace the Excelsior as the workhorse of the fleet and face of Starfleet, and she is one monster of a Heavy Cruiser (the closest real-world analogue would be the Alaska class Large Cruiser built at the end of WWII - both aren't large enough or powerful enough to be classed as Battlecruisers, but they're considerably larger and more powerful than other Heavy Cruisers). The Sovereign also has a critical advantage over the Galaxy: Speed. We know the Intrepid and Prometheus classes are the two fastest ships in the fleet (the Intrepid can sprint a little faster, but her nacelles can't bear the strain, and the Prometheus has an insanely high sustainable warp speed), but the Sovereign probably comes in at a close third place. I'm fairly confident, with her over-sized warp core and over-sized nacelles, that the Sovereign would be able to sprint faster than a GCS and sustain a faster warp speed than a GCS. Given that the intent is to get home as quickly as possible, and that there will be many situations where even a GCS isn't enough to win, the Sovereign's ability to run faster may well be the critical advantage.