A task that kills the EABackgroundService process when the launcher is fully closed

Kill_EABackgroundService.xml

<?xml version="1.0" encoding="UTF-16"?>
<Task version="1.4" xmlns="http://schemas.microsoft.com/windows/2004/02/mit/task">
  <RegistrationInfo>
    <Author>Katzenwerfer</Author>
    <Description>Kills the EABackgroundService process after the launcher finishes its exit process.</Description>
    <URI>\Katzenwerfer\Kill_EABackgroundService</URI>
  </RegistrationInfo>
  <Triggers>
    <EventTrigger>
      <Enabled>true</Enabled>
      <Subscription>&lt;QueryList&gt;&lt;Query Id="0" Path="Security"&gt;&lt;Select Path="Security"&gt;*[System[Provider[@Name='Microsoft-Windows-Security-Auditing'] and Task = 13313 and (EventID=4689)]] and *[EventData[Data[@Name='ProcessName'] and (Data='C:\Program Files\Electronic Arts\EA Desktop\EA Desktop\EADesktop.exe')]]
    &lt;/Select&gt;&lt;/Query&gt;&lt;/QueryList&gt;</Subscription>
    </EventTrigger>
  </Triggers>
  <Principals>
    <Principal id="Author">
      <LogonType>S4U</LogonType>
      <RunLevel>HighestAvailable</RunLevel>
    </Principal>
  </Principals>
  <Settings>
    <MultipleInstancesPolicy>StopExisting</MultipleInstancesPolicy>
    <DisallowStartIfOnBatteries>false</DisallowStartIfOnBatteries>
    <StopIfGoingOnBatteries>false</StopIfGoingOnBatteries>
    <AllowHardTerminate>true</AllowHardTerminate>
    <StartWhenAvailable>true</StartWhenAvailable>
    <RunOnlyIfNetworkAvailable>false</RunOnlyIfNetworkAvailable>
    <IdleSettings>
      <StopOnIdleEnd>false</StopOnIdleEnd>
      <RestartOnIdle>false</RestartOnIdle>
    </IdleSettings>
    <AllowStartOnDemand>true</AllowStartOnDemand>
    <Enabled>true</Enabled>
    <Hidden>false</Hidden>
    <RunOnlyIfIdle>false</RunOnlyIfIdle>
    <DisallowStartOnRemoteAppSession>false</DisallowStartOnRemoteAppSession>
    <UseUnifiedSchedulingEngine>true</UseUnifiedSchedulingEngine>
    <WakeToRun>false</WakeToRun>
    <ExecutionTimeLimit>PT1H</ExecutionTimeLimit>
    <Priority>7</Priority>
    <RestartOnFailure>
      <Interval>PT1M</Interval>
      <Count>5</Count>
    </RestartOnFailure>
  </Settings>
  <Actions Context="Author">
    <Exec>
      <Command>C:\Windows\System32\WindowsPowerShell\v1.0\powershell.exe</Command>
      <Arguments>-NoLogo -NoProfile -NonInteractive -WindowStyle Hidden -Command "Stop-Process -Name 'EABackgroundService' -Force"</Arguments>
      <WorkingDirectory>C:/</WorkingDirectory>
    </Exec>
  </Actions>
</Task>

If you want the PowerShell window to be completely hidden, you need to configure the task to Run wheter the user is logged on or not.
If you'd like, you can tick the Do not store password property. It will still run just fine.

Do note that you need to enable the Audit process tracking policy for it to work.
It should be located in \Local Computer Policy\Computer Policy\Windows Settings\Security Settings\Audit Policy\