How to send mail from a restricted SQL Server
So, I used the below script. 

Prerequisites are that the refresh script will write the log to a shared location accessible from a Prod Server.

$date = Get-Date -Format d
$dir = "\\xxxxx\g$\QARefreshLogs"
$recipient = "xxxxxx" 
$smtpserver = "xxxxxxx" 
$smtpport = "xx"
$smtp = new-object Net.Mail.SmtpClient($smtpServer)
$msg = new-object Net.Mail.MailMessage 
$msg.From = "xxxxxxxxxx"
$msg.To.Add($recipient)

---Retrieving the latest file 

$latest = Get-ChildItem -Path $dir | Sort-Object LastAccessTime -Descending | Where-Object {$_.Name -ilike "&lt;db_name&gt;_refresh_log_*"} | Select-Object -First 1
$LatestRefreshLog = $latest.Name
$LatestRefreshLogDate = $latest.LastWriteTime.ToString('M/dd/yyyy')

--Validating if the log file generated today?

if (($LatestRefreshLogDate) -notmatch $date)
     {
      #Write-Host "WARNING: There is no today's restore log file!"
      $msg.Subject = "SQLServer QA Refresh Status" 
      $msg.Body = "QA MSSQL DB refresh is having issues.Please validate."
      $smtp.Send($msg)
     }
     else
     {

--Validating if refresh is success or not?

      If(Get-Content -Path $dir\$LatestRefreshLog | %{$_ -match "RESTORE DATABASE successfully processed"})
      {
       #Write-Host "The latest restore log file is $LatestRefreshLog"
       $msg.Subject = "SQLServer QA Refresh Status" 
       $msg.Body = "QA DB is refreshed successfully."
       $smtp.Send($msg)
      }
      else
      { 
       #Write-Host "The latest restore log file is $LatestRefreshLog"
       $msg.Subject = "SQLServer QA Refresh Status" 
       $msg.Body = "QA DB refresh is having issues.Restore failed."
       $smtp.Send($msg)
     } }


This may not be an optimal script. It served my purpose....thanks for all your inputs. This script could be helpful to others, have a PoSh time!!!