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.
You have two big issues here.
First is that you have a serious risk of venting sewer gasses into your home. This is why drain vents open above the roof line or use one-way air admittance valves. This is likely to be unpleasant and possibly unhealthy.
Second is that a drain vent is typically too small and too moist to properly vent a bathroom fan. Your fan is probably designed to use a 4" duct, and instead it's going through a 1.5" diameter pipe. And if any lint/dust gets pushed out by your fan, it risks accumulating on the sides of the vent pipe, potentially blocking the pipe and slowing both your drainage and ventilation.
Beyond those functional problems, you'll also have a pretty visible issue that anyone inspecting your attic will see. You should do this the right way: add a properly-sized duct line and a roof vent for ventilation, and re-seal your drain vent pipe.
Best Answer
From what you describe it seems unlikely that the vent fan is actually installed to blow backwards. So it would be much more likely that the trouble you are encountering could be one of:
In all three instances there can be blockage of the airflow that the fan is trying to expel from the room. For a significant enough blockage some of the air flow will try to return back into the room around the periphery of the fan blades. This is likely what you are noticing when you held the paper up to the vent grille. It is often the case that a fan trying to blow into a blockage will generate a lot more noise than one that has a free flowing path for the air that the fan is moving.
The fix is going to require careful inspection of the whole vent system from the fan housing all the way through the vent pipe and out to the place where the pipe terminates the roof or wall of the building.