Does a bathroom exhaust fan have to vent to the outside

Check the directions/specifications of your exhaust fan, it should say if it is rated to have insulation next to or on top of it.

It would probably be a good idea to seal the vapour barrier back around the fan.

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.