I've been looking at this for about 2 hours now and I think that there is a way to do this, but it requires some more insight on how windows handles iexplore.exe for PID...
I have a working version of a batch file I wrote that will get you what you want, BUT only if its the FIRST AND ONLY Internet Explorer Window open.
For some reason I can't get the PID to change when I open new browsers, but I can get results if there is no window open (obviously because there is no PID)
Anyhow, this is what I have... you should be able to run this on your system and it will tell you that there are no differences and it might actually produce results if your default browser is Firefox or Chrome or something... just need to make the changes to what I'm providing.
@echo off
IF EXIST c:\temp\pshell.txt del c:\temp\pshell.txt
IF EXIST C:\temp\PID1.txt del C:\temp\PID1.txt
IF EXIST C:\temp\PID2.txt del C:\temp\PID2.txt
IF EXIST C:\temp\PowerFormat.txt del C:\temp\PowerFormat.txt
powershell.exe Get-Process iexplore>C:\temp\pshell.txt
FOR /F "skip=3 tokens=7 delims= " %%1 IN ( c:\temp\pshell.txt ) DO @echo %%1>> C:\temp\PID1.txt
start "title" "www.google.com"
powershell.exe Get-Process iexplore>C:\temp\pshell.txt
FOR /F "skip=3 tokens=7 delims= " %%2 IN ( c:\temp\pshell.txt ) DO @echo %%2>> C:\temp\PID2.txt
FC /L c:\temp\pid1.txt c:\temp\pid2.txt> C:\temp\FileComparison.txt
FOR /F "tokens=7 delims=" %%3 IN (c:\temp\FileComparison.txt) DO @echo %%3>C:\temp\DiffPID.txt
FINDSTR "FC: no differences encountered" c:\temp\FileComparison.txt
IF '%ERRORLEVEL%'=='0' del C:\temp\FileComparison.txt & echo.No PID Found
IF NOT '%ERRORLEVEL%'=='0' type c:\temp\FileComparison.txt
pause
exit
Let me know if this helps...