In order to get the Targus docks to work with the latest Catalina update we had to use the following steps.
1) Install latest DisplayLink Beta driver
https://www.displaylink.com/downloads/file?id=1498
2) Plug in the AC power adapter that came with the Mac
3) open the Security > Privacy > Screen Recording (Notice there is nothing to click)
4) Plug in the docking station (which also has its own power)
5) The DisplayLinkUserAgent checkbox will now show up so you can check it
6) Reboot and remove the AC Power Adapter
Once these steps were completed the docking station continued to function, until rebooted and the DisplayLink Manager software was not running.
Opening DisplayLink Manager and going to the settings to select "Launch at Startup" resolved this issue as well.