Получите последний файл и сравните его имя с остальными файлами.

Немного неясно, хотите ли вы использовать дату, которая является частью имени файла в перемещенном файле, или текущую дату. Вы также не говорите, в каком формате должна быть эта дата при использовании ее в новом имени файла.

См. комментарии в коде ниже для альтернатив на дату

$ParentFolder = 'Z:\WhereTheFilesAre'
$OutputFolder = 'X:\TheSestinationFolder'

Get-ChildItem -Path $ParentFolder -Filter '*.png' -File | Group-Object {$_.BaseName.Split("(")[0].TrimEnd()} | 
Where-Object { $_.Count -gt 1 } | ForEach-Object {
    # get the latest file
    $newestFile = $_.Group | Sort-Object LastWriteTime -Descending | Select-Object -First 1
    if ($newestFile.BaseName -match '^(.+)\s+(\d{2}_\d{2}_\d{4}\s+\d{2}_\d{2}_\d{2}\s+[ap]m).*$') {
        $screentitle      = $matches[1]
        $dateFromFileName = $matches[2]  # the date from the filename unaltered like '11_21_2022 10_59_21 AM'

        # if you want the date from the file name formatted like 'yyyy_MM_dd HH-mm-ss'
        # $date = [datetime]::ParseExact($dateFromFileName, 'MM_dd_yyyy h_mm_ss tt', [cultureinfo]::InvariantCulture)
        # $dateFromFileName = '{0:yyyy_MM_dd HH-mm-ss}' -f $date

        # create the new filename
        $NewFileName = '{0}_[{1}]{2}' -f $dateFromFileName, $screenTitle, $newestFile.Extension

        # Move the file with a new name to the destination
        Write-Host "Moving file '$($newestFile.Name)' as '$NewFileName'"
        $newestFile | Move-Item -Destination (Join-Path -Path $OutputFolder -ChildPath $NewFileName)

        # what do you want to do with the rest of the files that were older? Delete them?
        # $_.Group | Sort-Object LastWriteTime -Descending | Select-Object -Skip 1 | Remove-Item
    }
}

Согласно вашему комментарию, ниже обновленного ответа:

$ParentFolder = 'Z:\WhereTheFilesAre'
$OutputFolder = 'X:\TheSestinationFolder'

Get-ChildItem -Path $ParentFolder -Filter '*.png' -File | 
Where-Object { $_.BaseName -match '^(.+)\s+(\d{2}_\d{2}_\d{4}\s+\d{2}_\d{2}_\d{2}\s+[ap]m).*$' } |  # filter more specific
Group-Object { $_.BaseName -replace '\s+\d{2}_\d{2}_\d{4}\s+\d{2}_\d{2}_\d{2}\s+[ap]m.*$' } |       # group on screen title
Where-Object { $_.Count -gt 1 } | ForEach-Object {
    # get the latest file
    $newestFile   = $_.Group | Sort-Object LastWriteTime -Descending | Select-Object -First 1
    $screentitle  = $_.Name

    # scenario 1: Take the date from the file name
    $dateFromFileName = $newestFile.BaseName -replace '^.+\s+(\d{2}_\d{2}_\d{4}\s+\d{2}_\d{2}_\d{2}\s+[ap]m).*$', '$1'
    # parse the date from the file name
    $date = [datetime]::ParseExact($dateFromFileName, 'MM_dd_yyyy h_mm_ss tt', [cultureinfo]::InvariantCulture)
    # create the new filename
    $NewFileName = '{0:yyyyMMddHHmmss}_[{1}]{2}' -f $date, $screenTitle, $newestFile.Extension

    # scenario 2: Use the LastWriteTime property from the newest file (much easier)
    # $NewFileName = '{0:yyyyMMddHHmmss}_[{1}]{2}' -f $newestFile.LastWriteTime, $screenTitle, $newestFile.Extension

    # Move the file with a new name to the destination
    Write-Host "Moving file '$($newestFile.Name)' as '$NewFileName'"
    $newestFile | Move-Item -Destination (Join-Path -Path $OutputFolder -ChildPath $NewFileName)

    # what do you want to do with the rest of the files that were older? Delete them?
    # $_.Group | Sort-Object LastWriteTime -Descending | Select-Object -Skip 1 | Remove-Item
}