If you make a button which runs
Copy WHENEXISTS=renameold
then it will automatically rename existing files without prompting you. The renamed files will have "(original)" (with a number if they're the second such file) appended to their names. As far as I know there isn't a way to make the renamed files use a different naming format (e.g. a date).
If all the files are in one directory then you could use select all files in the source and then run Select SOURCETODEST so that any files with the same names in the destination are also selected. You could then run a rename button which renames those files to include the current date/time, and then do a normal copy of the new files. I suspect the files aren't all in the same directory, though, so this may not be that useful.
You could probably use the Synchronize tool to get a list of those files which need updating, and then select them and run the date-rename button instead of going ahead with the sync. Still takes a few steps, though.
If the dates are just a nice-to-have and automation is more important then I'd say go with the Copy WHENEXISTS=renameold method.
If you do want to rename files with the current date/time then this post might be helpful: