Bathroom is under a shallow slope roof on a bathroom bumpout on a 1940's cape cod. Owner found the bathroom fan is not vented at all, which explains why it wasn't doing much. Without access from above, and with concerns about insulation and drafts, would you install a vent fan working within in the room only, or insulate and close it up since there is a window in the room anyway. The joists are running toward the window and I think they are 2x10's.

Do you think that's enough room to install from below, including insulating tightly, stopping heat loss, etc.? Do they make IC (insulating contact) vent fans?

Final issue is that the fascia is only 2x6, leaving little room for tucking an exhaust duct right under the gutter.

Thanks.