Dear CyberShot, thank you for your input, please note that you need to mention full path in the Start-Process command, otherwise you will get the below message:

Here is a few changes in your command for better result, also I replaced the direct static link with the dynamic link for LsAgent which works fine under PowerShell:
Invoke-WebRequest -Uri "https://content.lansweeper.com/lsagent-windows/" -OutFile "C:\Windows\TEMP\LsAgent-windows.exe"; Start-Process -FilePath "C:\Windows\TEMP\LsAgent-windows.exe" -ArgumentList "--mode unattended --server YOUR_LANSWEEPER_FQDN --agentkey YOUR_LANSWEEPER_AGENT_CLOUDRELAY_KEY" -Wait
Although the downloaded file is working correctly.
Omar S. Bahabri