Venting bathroom through dryer?
I was able to do something simular. But my second bathroom is on the same level as the dryer. I did some testing first, to see how it would work out.
I only hooked up one backdraft damper on the dryer side to a Y fitting. Its important to install any backdraft fitting parallel to the flow. I let the dryer run with heat. After 10 minutes, I felt the metal exhaust duct about 6 feet away from the Y fitting, that is connected to the bathroom exhaust and it was warm. This tells me that heat, moisture and possibly some lint will work its way back to the bathroom. Then I added another backdraft damper to the bathroom ductwork in parallel to the flow. This created enough resistance to make all of the dryer flow go out its intended exhaust. When I felt the bathroom duct again, it was cool to the touch.
When you install this, do not use any type of foil or vinyl duct to either the dryer or the bathroom fan. Use smooth metal duct only because foil and vinyl are not as fire resistant as metal duct. Exception: If you were installing an exhaust from a bathroom directly to the outside, then it is permissable. But they also cause more resistance to the flow then smooth metal duct.
They are also not approved for use on dryers.
One last thing: Look for the dampers that have a single flapper valve rather then the 3 or 4 flappers that I see now in the home centers. The single flapper type has less resistance to the flow. You might find these in plumbing shops. Using your finger, check to see if the valve will open completly parallel to the flow. There may be a small bit of plastic actiing as a stopper. Use a file to file it smooth, if you have to. You will probally have to remove the valve from the assembly, but it is easy enough to do.
When your ready to install, make sure you have the damper flow facing towards the exhaust. Mistakes can happen. I put arrow marks on my dampers before I installed them. Install the dampers in a way that will make it easy to remove them in case you need to do future maintainace around them. Then test your installation.