Unless you know for certain the exit location of the bathroom fan, that option should be rejected. Our bathroom fans exit into the attic, which damages the insulation with moisture. We have them disconnected for that reason. Not good for ventilating a 3D printer either!

You have a couple of options for constructing a ventilation/filtration system. To prevent localized airflow from compromising the print quality, you'll want to have large volumes of air moving slowly compared to fast moving small volume. This means a large enclosure for the printer, which may not be practical.

Consider instead that you can build a ducting system to the window similar to the manner you suggest, but the fan should be large, perhaps floor-fan or window-fan sized. A common size in the US is a 20" box fan. You could place the fan with the exhaust aimed to the window, but placed close to the printer. It will "collect" air from around the area of the printer without creating fast moving cooling on your part. On the output side of the fan, plastic sheeting tapering to a box to match the window opening will direct the blown air, compressing it slightly before it exits the window. If you can match the window duct to the fan size or make it larger, all the better.

Some people will create a box on the exit of the fan, and build filters into the walls of the box. Carbon filters (expensive) are pretty effective in removing volatile organic compounds (VOCs) if the filter is so marked on the packaging.

Some people will also create an enclosure, either exhausting to the window or to a filter box, but keep the air flow to a lower figure, in order to prevent the inappropriate cooling of the printed part.