Quite simple via AppleScript.
Paste the script into a new Script Editor document, or in a 'Run AppleScript' action in Automator.
First pass: super simple - look for any file whose name ends in 'DATE.png' and rename it:
tell application "Finder"
try
set a_file to first file of (path to desktop) whose name ends with "DATE.png"
set name of a_file to "DATE.png"
end try
end tell
A slightly more robust version looks for specific filenames (otherwise "some other date.png' would also match above).
This one uses a list of viable file names and changes the first one it finds:
set filenames to {"Monday DATE.png", "Tuesday DATE.png", "Wednesday DATE.png", "Thursday DATE.png", "Friday DATE.png", "Saturday DATE.png", "Sunday DATE.png"}
tell application "Finder"
try
set a_file to first file of (path to desktop) whose name is in filenames
set name of a_file to "DATE.png"
end try
end tell
In both cases the 'try/end try' statements silently catch issues where no matching file can be found.