I have the same issue. The screen record permissions don't stick. I can go for some time before they need to be reset. Sometimes it can happen many days in a row and other times not at all.
MacOs 12.3.1
MacBook Pro (13-inch, M1, 2020)
DisplayLink Manager 1.6.1 (build 59.135)
I have just gone into the screen recording privacy settings and uncheck, then quit and reopen, then re-check then quit and reopen and everything works again.
Frustrating.