Hello Ed,
This is not a problem with DisplayLink technology or even your LAPDOCK.
DisplayLink chip which the LAPDOCK manufacturer has decided to use in that product, like your PS2/USB converter, is connected to the LAPDOCK internal USB dock.
The DisplayLink chip or driver has zero control or visibility of third party traffic.
You have found the issue yourself, even when connecting the adapter to your laptop directly. The PS/2 adapter is not recognised properly.
I see that, in Device Manager, the device recognised as Barcode Reader is Hardware ID "USB\VID_13BA&PID_0018" and it has zero driver assigned to it. It also has no manufacturer, service or driver assigned to it.
The fact it appears as Barcode Reader is because the adapter manufacturer reused a Hardware ID already allocated to something else. I can see the adapter has,
in the past, been allocated "USB\VID_13BA&PID_0017", but this is not something you can change yourself.
Your computer just doesn't know what to do with that adapter.
I recommend you contact the adapter manufacturer and ask them how it's supposed to work on Windows 10. They might give you either a specific driver or a procedure on how to force the keyboard class driver to the device.
Kind regards,
Alban