your approach is working perfectly for me:
$ whoami
test
$ echo $SHELL
/bin/bash
$ command="cd $directory && sh backupSQLToS3.sh"
$ job="15 1 */2 * * $command"
$ crontab -l
$ cat <(fgrep -i -v "$command" <(crontab -u test -l)) <(echo "$job") | crontab -u test -
$ crontab -l
15 1 */2 * * cd && sh backupSQLToS3.sh
I missed to set the "directory" variable but your code works fine for me.