It says a lot about support when the issue is passed along to the Senior Engineer and their reply is as follows:
"Thank you for the information.
It looks like we are not getting the "LastSuccessSync" from PowerShell on the scanning server. It is not clear to us why this date is not fetched while all other Exchange information is correct. It is not a known problem to us."
And from there they proceed to blame the OS or an incompatibility between the versions of PowerShell instead of looking into what the cause of the problem is.
Side note: I ran the commands in PowerShell from the Lansweeper hosting server to the Exchange server and the query came back with all of the information, including LastSuccessSync. That tells me that it is a Lansweeper problem...