Something like this would probably work (untested):

# Get the day of month as a number
day=$(date +%-d)

# Determine the correct suffix
case $day in
    1|21|31) suffix="st";;
    2|22) suffix="nd";;
    3|23) suffix="rd";;
    *) suffix="th";;
esac

# Print the date with the ordinal suffix
formatted_date=$(date +"%B %-d$suffix, %Y")