We’d like to be able to backup all of the modified, added, and conflicted files in our working copy.

svn status | grep -v ^[?D] | sed -e 's/^........\(.*\)$/.\/\1/g' | sed -e 's/\\/\//g' | tar czvf backup-`date +%Y_%m_%d`.tgz --files-from -

The grep may have to be adjusted a bit depending upon the state of your sandbox.

The second sed is due to the use of a native Windows version of subversion in Cygwin.

This post about removing unversioned files may also help. It provides some rationale for the first sed command.