How can I make OS X's bash expand $PWD to be case sensitive?
You have two choices really: convert $PWD to a consistent case before comparing to your other string (which should also have consistent case). Or drop case-sensitive comparisons.

Converting it to lower case before comparing:

#!/bin/bash
if [ `echo $PWD | tr [:upper:] [:lower:]` != /users/myuser/documents/backup ]
then
    echo "Wrong path: $PWD"
    exit 0
fi


Dropping case sensitive comparisons:

#!/bin/bash
shopt -s nocasematch
if [[ "$PWD" != /Users/myuser/Documents/Backup ]]
then
    echo "Wrong path: $PWD"
    exit 0
fi