We have dry cold winters as well, -35C, but not arctic. We exhaust bath moisture outside, but our fuel costs are pretty cheap. It would be worth estimating just how much extra heating costs are incurred by exhausting outside, it may not be that bad. Or a good argument why not to go outside. Excessive fan use can be limited by installing timer or occupancy switches.
I suspect the air exchanger doesn't work well because it ices up. It's worth investigating the cause though, in case it is just a clogged filter. A direct exhaust ices up outside around the outlet instead of inside the exchanger. The ice build up can be dramatic after many weeks below freezing and no sun. Another reason against outside exhaust besides being inefficient.
I don't think exhausting from one small space to another is a good idea. If your idea is to work, it would need to exhaust into the largest space. Perhaps above a heat source. I think some will find this objectionable, it's a bit unsanitary sounding. Sort of like using your roommate's damp bath towel. Nothing really wrong with it, yet quite unappealing. Especially if someone used the fan to remove odors instead of moisture! Eww. It's also possible that despite the dryness, heavy moisture load may just be too much to be dispersed within the dwelling.
You should focus on a workable air exchanger solution first. It is the only solution that is both energy efficient and fully sanitary. A direct exhaust may turn out to not be overly expensive, but it's still inefficient and should be avoided. Moving the moisture around inside may work, but should be the last resort, it just sounds a bit unsanitary, and it may not even work in all cases.
The fan may be able to be salvaged, only inspection and testing after it has been dried out can determine this. All the components should be removable from the metal box that is fastened to the ceiling joists. Be sure to turn off power at the breaker before doing any electrical work, though many fans simply unplug from the containing box. Just to be safe. Power can be restored once everything is out and any loose wires capped with wire nuts, one to each wire.
Dry out the components and inspect for mold. Any mold on non-electrical components can be cleaned with a bleach and water solution. If the motor appears moldy or water damaged, it's best to replace the fan.
Assess the limits of water damage in the ceiling, it can be much more extensive than is apparent from the painted side. All water damaged gypsum board or plaster must be replaced. From here on, I'll assume this is a gypsum board ceiling. Locate ceiling joists adjacent to the damaged area. Cut out a rectangular area slightly larger than the damage with edges falling on the center of the joists. You can cut between joists with a handsaw, but you must first be sure you are not damaging any wiring. The wet board can be pulled out by hand, allowing enough access to determine if any wiring is in the way.
The cuts along ceiling joists can be done with a utility knife. Be warned you will likely hit screws or nails along the way, remove them as you find them. Before doing this part, try to determine if the water damage extends into the next joist bay by inspecting from above, if possible.
Once you have removed all wet material, cut a new piece of gypsum board to fit the opening. Screw in place along each joist at each edge and every couple feet if it gets that big. Also screw in place the original board's new edge where nails or screws were removed. The technique for patching the joints is the same as for taping and mudding end joints of new gypsum board, there's many references on this process. It's not much more than hard core spackling.
Once the joints have been blended out enough to appear flat, prime and paint to match the remaining ceiling. This will all take some time, mainly waiting for the joint compound layers and paint coats to dry. The only hard work is cutting out a rectangular shape.
Best Answer
Do you want to have to replace it again if there's ever another leak? If yes, then use another inexpensive paper-and-gypsum-based product, such as drywall, green board, etc. If no, then use cementboard. Maybe overkill, but for a small bathroom the uncharge won't be more than $40 and that stuff can get wet till the cows come home and not complain about it! Once it's up, you can texture it, skim coat it, paint it, or whatever you were going to do to the drywall that you were considering.