This should do it:
@echo off
color 02
cd /D C:\Windows\System32
timeout -t 1
SET TempFile="%Temp%\%RANDOM%.txt"
:WaitSession
REM Fetch the current process list. Store in a temp file for easy searching.
TASKLIST > %TempFile%
REM Reset the process "flag" variable.
SET "Process="
REM Check for a process with the target name by searching the task list for the target process name.
REM If output is returned, it will be put into the Process "flag" variable.
FOR /F "usebackq tokens=1 delims=-" %%A IN (`FINDSTR /I "MightyQuest.exe" %TempFile%`) DO SET Process=%%A
REM If nothing was returned, it means the process isn't running.
IF "%Process%"=="" GOTO KillSession
ECHO Process is still running.
REM Wait and then try again.
TIMEOUT /T 20
GOTO WaitSession
:KillSession
taskkill /f /im PublicLauncher.exe
taskkill /f /im AwesomiumProcess.exe
REM Cleanup.
DEL %TempFile% > nul
The idea here is you keep polling the active task list and when the target process is found, you delay for a few seconds and then try again.
Once it is not found in the task list, you jump to the KillSession
steps.