What you are looking for is called a QFN ZIF (zero insertion force) socket.
(A few manufacturers include Aprilog and Test sockets, but they only have square pined QFN ZIF sockets listed). It's likely that manufactures mostly produce square QFN sockets because square packages are much more common. You might reach out and see if there are custom modules avalible.
It's also likely that you wouldn't have to test every single component, and just spot test the reel\tray\tube that the components come in, or every X components (depending on how much time and resources you have). I'd probably check at least 3 or 4 in the same package. You also might want to consider the cost of rework and ask the assembly house how much rework is (and get a quote), because if you could test the part natively on the PCB you are installing these on, it might not be that much more costly to do rework then testing them in the first place.