My car has developed a clunk when downshifting into 2nd and 1st gear. One of the exhaust hangers broke a couple months ago and it has caused a slight vibration. Can the exhaust cause the transmission to clunk?
My car has an automatic transmission.
The only way a loose exhaust system could cause a clunking noise (or any "banging" noise) is if the exhaust pipes are physically contacting the body of the car. If the clunking noise you are referring to is "new", and is coming from the transmission as opposed to, say, a failed CV axle joint, then there is likely a worn part in the transmission.
With regard to DSG transmissions, reports of clunking noises are readily discoverable on the Internet however it is unclear whether there is any known systematic problem with that particular transmission.
At any rate, the exhaust is probably not causing the noise you are hearing and so there probably is a issue with your transmission or, if you are lucky, a problem with a simpler to work on part of the drivetrain namely the CV axle.
In order to avoid any unnecessary repairs, YourMechanic could set up a mobile visit and determine what the cause of the noise is.