If you have access to Powershell on both machines, you may be able to use Invoke-Command or Invoke-CommandAs to run the command
This admittedly might be more powerful than a regular .bat execution, but based on past experience, would be how in automation environments we've had our Jenkins instances run commands remotely on nodes from the Jenkins host server machine.
In Powershell, there's an Invoke-Command option, which since I last looked at it, appears to have added additional parameters for running effectively as another user on that remote machine (I'm more familiar with the Invoke-CommandAs additional installation component, so I'm not 100% of this particular solution.). That said, based on the documentation there, the following is an example I'm copying here in case that link dies, and for quick reference.
Invoke-Command -FilePath <Path_To_Script*> -ComputerName <ComputerName**>
If, however, you're hoping to use the Invoke-CommandAs approach, this works from at least Powershell version 3.0, as I understand.
That would then look like this:
Invoke-CommandAs -ComputerName '<ComputerName**>' -Credential $Credential -ScriptBlock { Get-Process }
Or:
Invoke-CommandAs -ComputerName '<ComputerName**>' -AsUser $Credential -ScriptBlock { <Path_To_Script*> }
* i.e. "c:\scripts\test.ps1" in their example, or in your case, "\Users\Bridge\Desktop\BatchFiles\StopStart.cmd" - just be sure to remove the angled brackets in the code examples above.
** NETBIOS name, IP address, or fully qualified domain name of one or more computers in a comma-separated list. If using an IP Address, you must also provided a Credential parameter, along the lines of, for example -Credential Domain01\User01