I've not tried, but looks like they have an MSI installer that might help. https://www.papercut.com/help/manuals/ng-mf/clienttool/user-client-install-windows/

Give this a go and review the output in activity log. 

start-process msiexec -argumentlist "/i pc-client-admin-deploy.msi AUTORUN=1 /qn /norestart /l*v $env:SystemRoot\temp\pc-client-install.log" -wait
Get-Content $env:SystemRoot\temp\pc-client-install.log

If you use the EXE you might be able to get the log like this…

$log = Get-Childitem "$env:LOCALAPPDATA\Temp\Setup log*"
Get-Content$log.fullname

Refencing this for log file location https://www.papercut.com/help/manuals/ng-mf/common/install-win/

I had tried something like that, however for some silly reason PaperCut requires the .msi and .exe to be ran FROM the server, it requires basically everything inside the “client” folder for a local install.  I have tried the scripting below, it does map the drive but nothing installs?

# Set network drive information
$DriveLetter = "P"  # Choose the drive letter you want to assign
$NetworkPath = "\\Papercut\PCClient\win"  # Replace with the network path
$ProgramPath = "$NetworkPath\pc-client.exe"  # Replace program.exe with the actual program name

# Map network drive
New-PSDrive -Name $DriveLetter -PSProvider FileSystem -Root $NetworkPath -Persist

# Check if the drive is successfully mapped
if (Test-Path "$DriveLetter`:") {
    # Run program from network drive
    Start-Process -FilePath $ProgramPath -Wait

    # Disconnect network drive
    Remove-PSDrive -Name $DriveLetter
} else {
    Write-Host "Failed to map network drive."

the last } is there I just didnt paste right lol 

Wondering if maybe you could archive the entire client folder to ZIP, then use PowerShell to Expand-Archive to a local path for installation.

Copy-Item client.zip $env:SystemRoot\temp\
Expand-Archive $env:SystemRoot\temp\client.zip

another option, might be to use Set-Location which is similar to typing cd C:\some-path\ in PowerShell

Set-Location "\\Papercut\PCClient\win"

Something like this? Its Frankenstein’d from what I used to install another PaperCut monitor.

$fileName = 'client.zip'
# Set the path to the client.zip file
$sPath = Split-Path $Script:MyInvocation.MyCommand.Path -Parent
$fPath = "$sPath\$fileName"

# Set the destination folder for unzipping
$destinationFolder = "C:\client"

# Unzip client.zip to C drive
Expand-Archive -Path $zipFilePath -DestinationPath $destinationFolder

# Run program from inside the new folder
Set-Location "$destinationFolder\PCClient\win"
Start-Process -FilePath .\client-local-install.exe

So this ended up sort of a 3 pronged deployment to get things running right. PaperCut really messed up an easy deployment needing the share etc. BUT, I dropped the zip down, and unzipped it. In a second policy I had it setup a log in task to ensure the shortcut exe worked. Pain but it works!

Thanks for the help @jack.smith! Insight got me going the right direction for sure!

My experience with PaperCut has been that the files stay in place, but the program registration disappears making it look like it is removed.
I no longer work in that environment but it might be worthwhile lodging a ticket with PaperCut as that client is a sinkhole of tech debt and probably needs a rewrite.