Is it possible to send error output to a variable in bash?

One option is to write the file where you do have permission:

cp -R SourceDir DestDir 2>${TMPDIR:-/tmp}/out.$$.err

Another option is to capture the error (and non-error) output in a variable:

output=$(cp -R SourceDir DestDir 2>&1)
echo "$output"

Note that the first variant creates a process-specific temporary file. The file name generated thus is not very secure; if you work in a hostile environment as root, you need to take more precautions about generating easily predicted file names like that one. For most users most of the time, it is sufficient, especially if the $TMPDIR environment variable points to some directory such as $HOME/tmp where other users cannot write. Using a name containing $$ ensures that if two people run the script at the same time, they do not accidentally interfere with each other's log file.

You should also take precautions to clean up the file if the process is interrupted. The way to do that is shown in this boilerplate (template):

tmp=${TMPDIR:-/tmp}/out.$$.err
trap "rm -f $tmp; exit 1" 0 1 2 3 13 15 # EXIT HUP INT QUIT PIPE TERM

# Command using temporary file
cp -R SourceDir DestDir 2>$tmp

# Analyse contents...

# Clean up
rm -f $tmp
trap 0
exit 0

You can exit with a non-zero status (exit $status); the key point is to remove the file and then cancel the 'trap on exit'. The signals HUP, INT, QUIT, PIPE and TERM are the most likely ones for a process to receive, and in all of those cases, the shell removes the temporary file and exits with a non-zero status (1 in the example).